Theorem ceqsralt 2882

Description: Restricted quantifier version of ceqsalt 2881. (Contributed by NM, 28-Feb-2013.) (Revised by Mario Carneiro, 10-Oct-2016.)

Assertion

Ref	Expression
ceqsralt	|- (( F/xps /\ 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 2619	. . . 4 |- (A.x e. B (x = A -> ph) <-> A.x(x e. B -> (x = A -> ph)))
2		eleq1 2413	. . . . . . . . 9 |- (x = A -> (x e. B <-> A e. B))
3	2	pm5.32ri 619	. . . . . . . 8 |- ((x e. B /\ x = A) <-> (A e. B /\ x = A))
4	3	imbi1i 315	. . . . . . 7 |- (((x e. B /\ x = A) -> ph) <-> ((A e. B /\ x = A) -> ph))
5		impexp 433	. . . . . . 7 |- (((x e. B /\ x = A) -> ph) <-> (x e. B -> (x = A -> ph)))
6		impexp 433	. . . . . . 7 |- (((A e. B /\ x = A) -> ph) <-> (A e. B -> (x = A -> ph)))
7	4, 5, 6	3bitr3i 266	. . . . . 6 |- ((x e. B -> (x = A -> ph)) <-> (A e. B -> (x = A -> ph)))
8	7	albii 1566	. . . . 5 |- (A.x(x e. B -> (x = A -> ph)) <-> A.x(A e. B -> (x = A -> ph)))
9	8	a1i 10	. . . 4 |- (( F/xps /\ 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 248	. . 3 |- (( F/xps /\ 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 1890	. . 3 |- (A.x(A e. B -> (x = A -> ph)) <-> (A e. B -> A.x(x = A -> ph)))
12	10, 11	syl6bb 252	. 2 |- (( F/xps /\ 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 325	. . 3 |- (A e. B -> (A.x(x = A -> ph) <-> (A e. B -> A.x(x = A -> ph))))
14	13	3ad2ant3 978	. 2 |- (( F/xps /\ A.x(x = A -> (ph <-> ps)) /\ A e. B) -> (A.x(x = A -> ph) <-> (A e. B -> A.x(x = A -> ph))))
15		ceqsalt 2881	. 2 |- (( F/xps /\ A.x(x = A -> (ph <-> ps)) /\ A e. B) -> (A.x(x = A -> ph) <-> ps))
16	12, 14, 15	3bitr2d 272	1 |- (( F/xps /\ A.x(x = A -> (ph <-> ps)) /\ A e. B) -> (A.x e. B (x = A -> ph) <-> ps))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358   /\ w3a 934  A.wal 1540   F/wnf 1544   = wceq 1642   e. wcel 1710  A.wral 2614

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-11 1746  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-3an 936  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-ral 2619  df-v 2861

This theorem is referenced by:  ceqsralv  2886