Theorem eqsex 834

Description: A useful equivalence related to substitution.

Hypotheses

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

Assertion

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

Proof of Theorem eqsex

Step	Hyp	Ref	Expression
1		exnal 721	. 2 |- (E.x -. (x = y -> -. ph) <-> -. A.x(x = y -> -. ph))
2		df-an 198	. . 3 |- ((x = y /\ ph) <-> -. (x = y -> -. ph))
3	2	biex 733	. 2 |- (E.x(x = y /\ ph) <-> E.x -. (x = y -> -. ph))
4		eqsex.1	. . . . 5 |- (ps -> A.xps)
5	4	hbne 699	. . . 4 |- (-. ps -> A.x -. ps)
6		eqsex.2	. . . . 5 |- (x = y -> (ph <-> ps))
7	6	negbid 463	. . . 4 |- (x = y -> (-. ph <-> -. ps))
8	5, 7	eqsal 833	. . 3 |- (A.x(x = y -> -. ph) <-> -. ps)
9	8	bicon2i 194	. 2 |- (ps <-> -. A.x(x = y -> -. ph))
10	1, 3, 9	3bitr4 158	1 |- (E.x(x = y /\ ph) <-> ps)

Syntax hints:  -. wn 1   -> wi 2   <-> wb 127   /\ wa 196  A.wal 672  E.wex 678   = weq 797

This theorem is referenced by:  cleljust 985  sb5 988

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-gen 677  ax-9 799

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679

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