Theorem equsex 1657

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 290	. . 3 |- ( ( x = y /\ ph ) -> ps )
4	1, 3	exlimih 1525	. 2 |- ( E. x ( x = y /\ ph ) -> ps )
5		a9e 1627	. . 3 |- E. x x = y
6		idd 21	. . . . 5 |- ( ps -> ( x = y -> x = y ) )
7	2	biimprcd 158	. . . . 5 |- ( ps -> ( x = y -> ph ) )
8	6, 7	jcad 301	. . . 4 |- ( ps -> ( x = y -> ( x = y /\ ph ) ) )
9	1, 8	eximdh 1543	. . 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 124	1 |- ( E. x ( x = y /\ ph ) <-> ps )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   A.wal 1283   = wceq 1285   E.wex 1422

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441  ax-i9 1464  ax-ial 1468

This theorem depends on definitions:  df-bi 115

This theorem is referenced by:  cbvexh  1679  sb56  1807  cleljust  1855  sb10f  1913