Theorem ax12a2-o

Description: Derive ax-c15 from a hypothesis in the form of ax-12 , without using ax-12 or ax-c15 . The hypothesis is weaker than ax-12 , with z both distinct from x and not occurring in ph . Thus, the hypothesis provides an alternate axiom that can be used in place of ax-12 , if we also have ax-c11 , which this proof uses. As theorem ax12 shows, the distinct variable conditions are optional. An open problem is whether we can derive this with ax-c11n instead of ax-c11 . (Contributed by NM, 2-Feb-2007) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Hypothesis	ax12a2-o.1	$$⊢x=z\to \left(\forall z\phi \to \forall x\left(x=z\to \phi \right)\right)$$
Assertion	ax12a2-o	$$⊢\neg \forall xx=y\to \left(x=y\to \left(\phi \to \forall x\left(x=y\to \phi \right)\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		ax12a2-o.1	$$⊢x=z\to \left(\forall z\phi \to \forall x\left(x=z\to \phi \right)\right)$$
2		ax-5	$$⊢\phi \to \forall z\phi$$
3	2 1	syl5	$$⊢x=z\to \left(\phi \to \forall x\left(x=z\to \phi \right)\right)$$
4	3	ax12v2-o	$$⊢\neg \forall xx=y\to \left(x=y\to \left(\phi \to \forall x\left(x=y\to \phi \right)\right)\right)$$