Theorem ax5eq

Description: Theorem to add distinct quantifier to atomic formula. (This theorem demonstrates the induction basis for ax-5 considered as a metatheorem. Do not use it for later proofs - use ax-5 instead, to avoid reference to the redundant axiom ax-c16 .) (Contributed by NM, 10-Jan-1993) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	ax5eq	$$⊢x=y\to \forall zx=y$$

Proof

Step	Hyp	Ref	Expression
1		ax-c9	$$⊢\neg \forall zz=x\to \left(\neg \forall zz=y\to \left(x=y\to \forall zx=y\right)\right)$$
2		ax-c16	$$⊢\forall zz=x\to \left(x=y\to \forall zx=y\right)$$
3		ax-c16	$$⊢\forall zz=y\to \left(x=y\to \forall zx=y\right)$$
4	1 2 3	pm2.61ii	$$⊢x=y\to \forall zx=y$$