Theorem ceqsralt 3229

Description: Restricted quantifier version of ceqsalt 3228. (Contributed by NM, 28-Feb-2013.) (Revised by Mario Carneiro, 10-Oct-2016.)

Assertion

Ref	Expression
ceqsralt	|- ( ( F/ x ps /\ A. x ( x = A -> ( ph <-> ps ) ) /\ A e. B ) -> ( A. x e. B ( x = A -> ph ) <-> ps ) )

Distinct variable groups:   x, A   x, B

Allowed substitution hints:   ph( x)   ps( x)

Proof of Theorem ceqsralt

Step	Hyp	Ref	Expression
1		df-ral 2917	. . . 4 |- ( A. x e. B ( x = A -> ph ) <-> A. x ( x e. B -> ( x = A -> ph ) ) )
2		eleq1 2689	. . . . . . . 8 |- ( x = A -> ( x e. B <-> A e. B ) )
3	2	pm5.32ri 670	. . . . . . 7 |- ( ( x e. B /\ x = A ) <-> ( A e. B /\ x = A ) )
4	3	imbi1i 339	. . . . . 6 |- ( ( ( x e. B /\ x = A ) -> ph ) <-> ( ( A e. B /\ x = A ) -> ph ) )
5		impexp 462	. . . . . 6 |- ( ( ( x e. B /\ x = A ) -> ph ) <-> ( x e. B -> ( x = A -> ph ) ) )
6		impexp 462	. . . . . 6 |- ( ( ( A e. B /\ x = A ) -> ph ) <-> ( A e. B -> ( x = A -> ph ) ) )
7	4, 5, 6	3bitr3i 290	. . . . 5 |- ( ( x e. B -> ( x = A -> ph ) ) <-> ( A e. B -> ( x = A -> ph ) ) )
8	7	albii 1747	. . . 4 |- ( A. x ( x e. B -> ( x = A -> ph ) ) <-> A. x ( A e. B -> ( x = A -> ph ) ) )
9		19.21v 1868	. . . 4 |- ( A. x ( A e. B -> ( x = A -> ph ) ) <-> ( A e. B -> A. x ( x = A -> ph ) ) )
10	1, 8, 9	3bitri 286	. . 3 |- ( A. x e. B ( x = A -> ph ) <-> ( A e. B -> A. x ( x = A -> ph ) ) )
11	10	a1i 11	. 2 |- ( ( F/ x ps /\ A. x ( x = A -> ( ph <-> ps ) ) /\ A e. B ) -> ( A. x e. B ( x = A -> ph ) <-> ( A e. B -> A. x ( x = A -> ph ) ) ) )
12		biimt 350	. . 3 |- ( A e. B -> ( A. x ( x = A -> ph ) <-> ( A e. B -> A. x ( x = A -> ph ) ) ) )
13	12	3ad2ant3 1084	. 2 |- ( ( F/ x ps /\ A. x ( x = A -> ( ph <-> ps ) ) /\ A e. B ) -> ( A. x ( x = A -> ph ) <-> ( A e. B -> A. x ( x = A -> ph ) ) ) )
14		ceqsalt 3228	. 2 |- ( ( F/ x ps /\ A. x ( x = A -> ( ph <-> ps ) ) /\ A e. B ) -> ( A. x ( x = A -> ph ) <-> ps ) )
15	11, 13, 14	3bitr2d 296	1 |- ( ( F/ x ps /\ A. x ( x = A -> ( ph <-> ps ) ) /\ A e. B ) -> ( A. x e. B ( x = A -> ph ) <-> ps ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   A.wal 1481   = wceq 1483   F/wnf 1708   e. wcel 1990   A.wral 2912

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-12 2047  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-ral 2917  df-v 3202

This theorem is referenced by:  ceqsralv  3234  cdleme32fva  35725