Theorem equsex 1513

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 1385	. 2 |- (E.x -. (x = y -> -. ph) <-> -. A.x(x = y -> -. ph))
2		df-an 242	. . 3 |- ((x = y /\ ph) <-> -. (x = y -> -. ph))
3	2	exbii 1398	. 2 |- (E.x(x = y /\ ph) <-> E.x -. (x = y -> -. ph))
4		equsex.1	. . . . 5 |- (ps -> A.xps)
5	4	hbn 1351	. . . 4 |- (-. ps -> A.x -. ps)
6		equsex.2	. . . . 5 |- (x = y -> (ph <-> ps))
7	6	notbid 673	. . . 4 |- (x = y -> (-. ph <-> -. ps))
8	5, 7	equsal 1511	. . 3 |- (A.x(x = y -> -. ph) <-> -. ps)
9	8	con2bii 238	. 2 |- (ps <-> -. A.x(x = y -> -. ph))
10	1, 3, 9	3bitr4i 200	1 |- (E.x(x = y /\ ph) <-> ps)

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   /\ wa 240  A.wal 1296   = wceq 1298  E.wex 1326

This theorem is referenced by:  sb56 1643  cleljust 1713  sb10f 1733  axsep 3437  iunfopabOLD 4543

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 1305  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481

This theorem depends on definitions:  df-bi 164  df-an 242  df-ex 1327

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