Theorem ceqsalt 2882

Description: Closed theorem version of ceqsalg 2884. (Contributed by NM, 28-Feb-2013.) (Revised by Mario Carneiro, 10-Oct-2016.)

Assertion

Ref	Expression
ceqsalt	|- (( F/xps /\ A.x(x = A -> (ph <-> ps)) /\ A e. V) -> (A.x(x = A -> ph) <-> ps))

Distinct variable group:   x,A

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

Proof of Theorem ceqsalt

Step	Hyp	Ref	Expression
1		elisset 2870	. . . 4 |- (A e. V -> E.x x = A)
2	1	3ad2ant3 978	. . 3 |- (( F/xps /\ A.x(x = A -> (ph <-> ps)) /\ A e. V) -> E.x x = A)
3		bi1 178	. . . . . . 7 |- ((ph <-> ps) -> (ph -> ps))
4	3	imim3i 55	. . . . . 6 |- ((x = A -> (ph <-> ps)) -> ((x = A -> ph) -> (x = A -> ps)))
5	4	al2imi 1561	. . . . 5 |- (A.x(x = A -> (ph <-> ps)) -> (A.x(x = A -> ph) -> A.x(x = A -> ps)))
6	5	3ad2ant2 977	. . . 4 |- (( F/xps /\ A.x(x = A -> (ph <-> ps)) /\ A e. V) -> (A.x(x = A -> ph) -> A.x(x = A -> ps)))
7		19.23t 1800	. . . . 5 |- ( F/xps -> (A.x(x = A -> ps) <-> (E.x x = A -> ps)))
8	7	3ad2ant1 976	. . . 4 |- (( F/xps /\ A.x(x = A -> (ph <-> ps)) /\ A e. V) -> (A.x(x = A -> ps) <-> (E.x x = A -> ps)))
9	6, 8	sylibd 205	. . 3 |- (( F/xps /\ A.x(x = A -> (ph <-> ps)) /\ A e. V) -> (A.x(x = A -> ph) -> (E.x x = A -> ps)))
10	2, 9	mpid 37	. 2 |- (( F/xps /\ A.x(x = A -> (ph <-> ps)) /\ A e. V) -> (A.x(x = A -> ph) -> ps))
11		bi2 189	. . . . . . 7 |- ((ph <-> ps) -> (ps -> ph))
12	11	imim2i 13	. . . . . 6 |- ((x = A -> (ph <-> ps)) -> (x = A -> (ps -> ph)))
13	12	com23 72	. . . . 5 |- ((x = A -> (ph <-> ps)) -> (ps -> (x = A -> ph)))
14	13	alimi 1559	. . . 4 |- (A.x(x = A -> (ph <-> ps)) -> A.x(ps -> (x = A -> ph)))
15	14	3ad2ant2 977	. . 3 |- (( F/xps /\ A.x(x = A -> (ph <-> ps)) /\ A e. V) -> A.x(ps -> (x = A -> ph)))
16		19.21t 1795	. . . 4 |- ( F/xps -> (A.x(ps -> (x = A -> ph)) <-> (ps -> A.x(x = A -> ph))))
17	16	3ad2ant1 976	. . 3 |- (( F/xps /\ A.x(x = A -> (ph <-> ps)) /\ A e. V) -> (A.x(ps -> (x = A -> ph)) <-> (ps -> A.x(x = A -> ph))))
18	15, 17	mpbid 201	. 2 |- (( F/xps /\ A.x(x = A -> (ph <-> ps)) /\ A e. V) -> (ps -> A.x(x = A -> ph)))
19	10, 18	impbid 183	1 |- (( F/xps /\ A.x(x = A -> (ph <-> ps)) /\ A e. V) -> (A.x(x = A -> ph) <-> ps))

Syntax hints:   -> wi 4   <-> wb 176   /\ w3a 934  A.wal 1540  E.wex 1541   F/wnf 1544   = wceq 1642   e. wcel 1710

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-v 2862

This theorem is referenced by:  ceqsralt  2883