Theorem ceqsralt 2716

Description: Restricted quantifier version of ceqsalt 2715. (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 2422	. . . 4 |- ( A. x e. B ( x = A -> ph ) <-> A. x ( x e. B -> ( x = A -> ph ) ) )
2		eleq1 2203	. . . . . . . . 9 |- ( x = A -> ( x e. B <-> A e. B ) )
3	2	pm5.32ri 451	. . . . . . . 8 |- ( ( x e. B /\ x = A ) <-> ( A e. B /\ x = A ) )
4	3	imbi1i 237	. . . . . . 7 |- ( ( ( x e. B /\ x = A ) -> ph ) <-> ( ( A e. B /\ x = A ) -> ph ) )
5		impexp 261	. . . . . . 7 |- ( ( ( x e. B /\ x = A ) -> ph ) <-> ( x e. B -> ( x = A -> ph ) ) )
6		impexp 261	. . . . . . 7 |- ( ( ( A e. B /\ x = A ) -> ph ) <-> ( A e. B -> ( x = A -> ph ) ) )
7	4, 5, 6	3bitr3i 209	. . . . . 6 |- ( ( x e. B -> ( x = A -> ph ) ) <-> ( A e. B -> ( x = A -> ph ) ) )
8	7	albii 1447	. . . . 5 |- ( A. x ( x e. B -> ( x = A -> ph ) ) <-> A. x ( A e. B -> ( x = A -> ph ) ) )
9	8	a1i 9	. . . 4 |- ( ( F/ x ps /\ A. x ( x = A -> ( ph <-> ps ) ) /\ A e. B ) -> ( A. x ( x e. B -> ( x = A -> ph ) ) <-> A. x ( A e. B -> ( x = A -> ph ) ) ) )
10	1, 9	syl5bb 191	. . 3 |- ( ( F/ x ps /\ A. x ( x = A -> ( ph <-> ps ) ) /\ A e. B ) -> ( A. x e. B ( x = A -> ph ) <-> A. x ( A e. B -> ( x = A -> ph ) ) ) )
11		19.21v 1846	. . 3 |- ( A. x ( A e. B -> ( x = A -> ph ) ) <-> ( A e. B -> A. x ( x = A -> ph ) ) )
12	10, 11	syl6bb 195	. 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 ) ) ) )
13		biimt 240	. . 3 |- ( A e. B -> ( A. x ( x = A -> ph ) <-> ( A e. B -> A. x ( x = A -> ph ) ) ) )
14	13	3ad2ant3 1005	. 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 ) ) ) )
15		ceqsalt 2715	. 2 |- ( ( F/ x ps /\ A. x ( x = A -> ( ph <-> ps ) ) /\ A e. B ) -> ( A. x ( x = A -> ph ) <-> ps ) )
16	12, 14, 15	3bitr2d 215	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   /\ wa 103   <-> wb 104   /\ w3a 963   A.wal 1330   = wceq 1332   F/wnf 1437   e. wcel 1481   A.wral 2417

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1424  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-3an 965  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-ral 2422  df-v 2691

This theorem is referenced by:  ceqsralv  2720