Theorem ax5el

Description: Theorem to add distinct quantifier to atomic formula. This theorem demonstrates the induction basis for ax-5 considered as a metatheorem.) (Contributed by NM, 22-Jun-1993) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	ax5el	$$⊢x\in y\to \forall zx\in y$$

Proof

Step	Hyp	Ref	Expression
1		ax-c14	$$⊢\neg \forall zz=x\to \left(\neg \forall zz=y\to \left(x\in y\to \forall zx\in y\right)\right)$$
2		ax-c16	$$⊢\forall zz=x\to \left(x\in y\to \forall zx\in y\right)$$
3		ax-c16	$$⊢\forall zz=y\to \left(x\in y\to \forall zx\in y\right)$$
4	1 2 3	pm2.61ii	$$⊢x\in y\to \forall zx\in y$$