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While the process of your proof is correct. The error occurred when the criteria was changed to something that's not used for marriage.
Say we are trying to prove size of A and B are the different. One can prove that by measure the area/volume which are the criteria of size. But conclusively proving that A and B have different color, doesn't do anything for the proof since color is not a measure of size.
I know what you are trying to say though. But bear in mind it's not just the liberals asking to make the leap that benefits their cause, the conservatives are asking for similar leaps. One can't jump both ways.
Here's the thing: that criteria doesn't define marriage; it limits who can join in the civil contract of marriage. It doesn't define the purpose and benefits of marriage; that is determined by state interest. The state interest in marriage is, in part, to create a set of rules and regulation for taxation, inheritance, health decision benefits, etc. Fiscally, marriage as a social contract serves to identify primary beneficiaries and caretakers for individuals. Many responsibilities exist in the dissolution of marriage as well, protecting two parties who enter such a social contract. This, too, is part of state interest of protecting citizens. Marriage also ensures that parents who have a kid while married are fiscally responsible. This is part of state interest by protecting the kid and the abandoned parent. It's also argued that marriage serves a state interest by providing value to society as an intermediate institution. Socially, there are rights and obligations of spouses to each other (though under family privacy this cannot be enforced). Historically the state interest was to preserve the nuclear family -- to keep a man tied down to a wife and kids. But with same-sex marriage and adoption being much more widespread, the same issues exist in creating a family unit. This is a simple argument of inclusion of gay and lesbian couples into the institution that is marriage.
You can try to argue that marriage is only a family issue, and thus only a state issue. But really marriage comes to play in the national realm too. In citizenship, if a non-citizen marries a same-sex citizen, what then? Do you defer to state inclusion criteria? If so, it's inconsistent. If you define inclusion criteria to include only man and woman, you are ignoring the fact that there are the same state interests that exist for both types of couples and families.