Theorem equsal 1184

Description: A useful equivalence related to substitution.

Hypotheses

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

Assertion

Ref	Expression
equsal	|- (A.x(x = y -> ph) <-> ps)

Proof of Theorem equsal

Step	Hyp	Ref	Expression
1		equsal.2	. . . . 5 |- (x = y -> (ph <-> ps))
2		equsal.1	. . . . . 6 |- (ps -> A.xps)
3	2	19.3 1063	. . . . 5 |- (A.xps <-> ps)
4	1, 3	syl6bbr 540	. . . 4 |- (x = y -> (ph <-> A.xps))
5	4	pm5.74i 586	. . 3 |- ((x = y -> ph) <-> (x = y -> A.xps))
6	5	albii 1031	. 2 |- (A.x(x = y -> ph) <-> A.x(x = y -> A.xps))
7		ax-1 4	. . . . 5 |- (A.xps -> (x = y -> A.xps))
8	7	a5i 1021	. . . 4 |- (A.xps -> A.x(x = y -> A.xps))
9	2, 8	syl 10	. . 3 |- (ps -> A.x(x = y -> A.xps))
10		ax-9o 1155	. . 3 |- (A.x(x = y -> A.xps) -> ps)
11	9, 10	impbii 155	. 2 |- (ps <-> A.x(x = y -> A.xps))
12	6, 11	bitr4i 174	1 |- (A.x(x = y -> ph) <-> ps)

Syntax hints:   -> wi 3   <-> wb 144  A.wal 986   = wceq 988

This theorem is referenced by:  equsex 1185  dvelimfALT 1186  dvelimALT 1386  fun11 3637

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 995  ax-4 1005  ax-5o 1007  ax-9o 1155

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

Copyright terms: Public domain