Metamath Proof Explorer


Theorem equs5av

Description: A property related to substitution that replaces the distinctor from equs5 to a disjoint variable condition. Version of equs5a with a disjoint variable condition, which does not require ax-13 . See also sbalex . (Contributed by NM, 2-Feb-2007) (Revised by Gino Giotto, 15-Dec-2023)

Ref Expression
Assertion equs5av
|- ( E. x ( x = y /\ A. y ph ) -> A. x ( x = y -> ph ) )

Proof

Step Hyp Ref Expression
1 nfa1
 |-  F/ x A. x ( x = y -> ph )
2 ax12v2
 |-  ( x = y -> ( ph -> A. x ( x = y -> ph ) ) )
3 2 spsd
 |-  ( x = y -> ( A. y ph -> A. x ( x = y -> ph ) ) )
4 3 imp
 |-  ( ( x = y /\ A. y ph ) -> A. x ( x = y -> ph ) )
5 1 4 exlimi
 |-  ( E. x ( x = y /\ A. y ph ) -> A. x ( x = y -> ph ) )