Theorem eqsal 833

Description: A useful equivalence related to substitution.

Hypotheses

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

Assertion

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

Proof of Theorem eqsal

Step	Hyp	Ref	Expression
1		eqsal.2	. . . . 5 |- (x = y -> (ph <-> ps))
2		eqsal.1	. . . . . 6 |- (ps -> A.xps)
3	2	19.3r 714	. . . . 5 |- (ps <-> A.xps)
4	1, 3	syl6bb 414	. . . 4 |- (x = y -> (ph <-> A.xps))
5	4	pm5.74i 443	. . 3 |- ((x = y -> ph) <-> (x = y -> A.xps))
6	5	bial 695	. 2 |- (A.x(x = y -> ph) <-> A.x(x = y -> A.xps))
7		ax-1 3	. . . . 5 |- (A.xps -> (x = y -> A.xps))
8	7	a5i 687	. . . 4 |- (A.xps -> A.x(x = y -> A.xps))
9	2, 8	syl 12	. . 3 |- (ps -> A.x(x = y -> A.xps))
10		ax9 807	. . 3 |- (A.x(x = y -> A.xps) -> ps)
11	9, 10	impbi 139	. 2 |- (ps <-> A.x(x = y -> A.xps))
12	6, 11	bitr4 154	1 |- (A.x(x = y -> ph) <-> ps)

Syntax hints:   -> wi 2   <-> wb 127  A.wal 672   = weq 797

This theorem is referenced by:  eqsex 834  ddelimf2 907  sb6 989

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

metamath.org