Theorem equs5e

Description: A property related to substitution that unlike equs5 does not require a distinctor antecedent. This proof uses ax12 , see equs5eALT for an alternative one using ax-12 but not ax13 . Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by NM, 2-Feb-2007) (Proof shortened by Wolf Lammen, 15-Jan-2018) (New usage is discouraged.)

	Ref	Expression
Assertion	equs5e	|- ( 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		ax12	|- ( x = y -> ( A. y E. y ph -> A. x ( x = y -> E. y ph ) ) )
3		hbe1	|- ( E. y ph -> A. y E. y ph )
4	3	19.23bi	|- ( ph -> A. y E. y ph )
5	2 4	impel	|- ( ( x = y /\ ph ) -> A. x ( x = y -> E. y ph ) )
6	1 5	exlimi	|- ( E. x ( x = y /\ ph ) -> A. x ( x = y -> E. y ph ) )