I think that you have made a mistake with regard to the role of argument. It is important to understand that a logical "proof" does not prove that the conclusion of an argument is true, but only that the conclusion is true when we assume that the premises are true, and it is quite clear that an argument does not prove its premises either. Therefore, the premises of an argument cannot support their conclusion, because the logical content of the conclusion is a subset of the logical content of the premises i.e. every deductive argument begs the question. The situation is even worse with regard to inductive arguments, because to the extent that the conclusion follows it does so circularly, and to the extent that it does not follow it is simply invalid. In other words, the premises of an argument cannot provide any support or good reason to think that their conclusion is true, a trivial consequence of elementary laws.
I also think that you should study the difference between an equation and a deduction. In short, every equation is also a deduction but every deduction is not also an equation i.e. the set of equations is a subset of the set of deductions. The difference is to do with symmetry i.e. the relation of equatibility is always symmetrical whereas deducibility can be asymmetrical. For example,
[indent]1. If A = B then B = A
2. If A |= B then B |= A[/indent]
The first statement is always true whereas the second is not always true. In fact, the second statement is true, if and only if, A is equal to B (remember, every equation is also a deduction). Therefore, while 1 + 1 = 2, the following argument is invalid:
Every dog is a mammal
& Every mammal is warm-blooded
= Every dog is warm-blooded
The invalidity is clear since equative arguments, unlike deductive arguments, must always be symmetrical i.e. the conclusion must follow from the premises and vice versa
. However, it is not the case that either of the premises is implied by the conclusion, and so your argument is a nonequative deductive argument. Therefore, the use of the equality relationship is invalid.