Theorem equsexhv 2124

Description: Version of equsexh 2295 with a dv condition, which does not require ax-13 2246. (Contributed by BJ, 31-May-2019.)

Hypotheses

Ref	Expression
equsalhw.1	|- ( ps -> A. x ps )
equsalhw.2	|- ( x = y -> ( ph <-> ps ) )

Assertion

Ref	Expression
equsexhv	|- ( E. x ( x = y /\ ph ) <-> ps )

Distinct variable group:   x, y

Allowed substitution hints:   ph( x, y)   ps( x, y)

Proof of Theorem equsexhv

Step	Hyp	Ref	Expression
1		equsalhw.1	. . 3 |- ( ps -> A. x ps )
2	1	nf5i 2024	. 2 |- F/ x ps
3		equsalhw.2	. 2 |- ( x = y -> ( ph <-> ps ) )
4	2, 3	equsexv 2109	1 |- ( E. x ( x = y /\ ph ) <-> ps )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   E.wex 1704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-nf 1710

This theorem is referenced by:  cleljustALT  2185