Theorem equsex 1189

Description: A useful equivalence related to substitution.

Hypotheses

Ref	Expression
equsex.1	|- (ps -> A.xps)
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		exnal 1074	. 2 |- (E.x -. (x = y -> -. ph) <-> -. A.x(x = y -> -. ph))
2		df-an 223	. . 3 |- ((x = y /\ ph) <-> -. (x = y -> -. ph))
3	2	exbii 1087	. 2 |- (E.x(x = y /\ ph) <-> E.x -. (x = y -> -. ph))
4		equsex.1	. . . . 5 |- (ps -> A.xps)
5	4	hbn 1040	. . . 4 |- (-. ps -> A.x -. ps)
6		equsex.2	. . . . 5 |- (x = y -> (ph <-> ps))
7	6	notbid 614	. . . 4 |- (x = y -> (-. ph <-> -. ps))
8	5, 7	equsal 1188	. . 3 |- (A.x(x = y -> -. ph) <-> -. ps)
9	8	con2bii 219	. 2 |- (ps <-> -. A.x(x = y -> -. ph))
10	1, 3, 9	3bitr4i 181	1 |- (E.x(x = y /\ ph) <-> ps)

Syntax hints:  -. wn 2   -> wi 3   <-> wb 144   /\ wa 221  A.wal 990   = wceq 992  E.wex 1016

This theorem is referenced by:  sb56 1304  cleljust 1366  sb10f 1381  axsep 2776  iunfopab 3719  elsb3NEWlem 12222  elsb4lem 12223  elsb2lem 12225

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 999  ax-4 1009  ax-5o 1011  ax-6o 1014  ax-9o 1159

This theorem depends on definitions:  df-bi 145  df-an 223  df-ex 1017

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