Metamath Proof Explorer


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 ) )