Theorem equsex 1721

Description: A useful equivalence related to substitution. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 3-Feb-2015.)

Hypotheses

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

Assertion

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

Proof of Theorem equsex

Step	Hyp	Ref	Expression
1		equsex.1	. . 3 |- ( ps -> A. x ps )
2		equsex.2	. . . 4 |- ( x = y -> ( ph <-> ps ) )
3	2	biimpa 294	. . 3 |- ( ( x = y /\ ph ) -> ps )
4	1, 3	exlimih 1586	. 2 |- ( E. x ( x = y /\ ph ) -> ps )
5		a9e 1689	. . 3 |- E. x x = y
6		idd 21	. . . . 5 |- ( ps -> ( x = y -> x = y ) )
7	2	biimprcd 159	. . . . 5 |- ( ps -> ( x = y -> ph ) )
8	6, 7	jcad 305	. . . 4 |- ( ps -> ( x = y -> ( x = y /\ ph ) ) )
9	1, 8	eximdh 1604	. . 3 |- ( ps -> ( E. x x = y -> E. x ( x = y /\ ph ) ) )
10	5, 9	mpi 15	. 2 |- ( ps -> E. x ( x = y /\ ph ) )
11	4, 10	impbii 125	1 |- ( E. x ( x = y /\ ph ) <-> ps )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   A.wal 1346   = wceq 1348   E.wex 1485

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503  ax-i9 1523  ax-ial 1527

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  cbvexv1  1745  cbvexh  1748  sb56  1878  sb10f  1988  cleljust  2147