Theorem equsexvw 1781

Description: Version of equsex 1780 with two disjoint variable conditions. (Contributed by BJ, 31-May-2019.) (Proof shortened by Wolf Lammen, 23-Oct-2023.)

Hypothesis

Ref	Expression
equsalvw.1	|- ( x = y -> ( ph <-> ps ) )

Assertion

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

Distinct variable groups:   x, y   ps, x

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

Proof of Theorem equsexvw

Step	Hyp	Ref	Expression
1		ax-17 1579	. 2 |- ( ps -> A. x ps )
2		equsalvw.1	. 2 |- ( x = y -> ( ph <-> ps ) )
3	1, 2	equsex 1780	1 |- ( E. x ( x = y /\ ph ) <-> ps )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   E.wex 1545

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  mapsnend  7092