Theorem equs5eALT

Description: Alternate proof of equs5e . Uses ax-12 but not ax-13 . (Contributed by NM, 2-Feb-2007) (Proof shortened by Wolf Lammen, 15-Jan-2018) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	equs5eALT	|- ( E. x ( x = y /\ ph ) -> A. x ( x = y -> E. y ph ) )

Proof

Step	Hyp	Ref	Expression
1		nfa1	|- F/ x A. x ( x = y -> E. y ph )
2		hbe1	|- ( E. y ph -> A. y E. y ph )
3	2	19.23bi	|- ( ph -> A. y E. y ph )
4		ax-12	|- ( x = y -> ( A. y E. y ph -> A. x ( x = y -> E. y ph ) ) )
5	3 4	syl5	|- ( x = y -> ( ph -> A. x ( x = y -> E. y ph ) ) )
6	5	imp	|- ( ( x = y /\ ph ) -> A. x ( x = y -> E. y ph ) )
7	1 6	exlimi	|- ( E. x ( x = y /\ ph ) -> A. x ( x = y -> E. y ph ) )