Theorem ceqsalt 2826

Description: Closed theorem version of ceqsalg 2828. (Contributed by NM, 28-Feb-2013.) (Revised by Mario Carneiro, 10-Oct-2016.)

Assertion

Ref	Expression
ceqsalt	|- ( ( F/ x ps /\ 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 2814	. . . 4 |- ( A e. V -> E. x x = A )
2	1	3ad2ant3 1044	. . 3 |- ( ( F/ x ps /\ A. x ( x = A -> ( ph <-> ps ) ) /\ A e. V ) -> E. x x = A )
3		biimp 118	. . . . . . 7 |- ( ( ph <-> ps ) -> ( ph -> ps ) )
4	3	imim3i 61	. . . . . 6 |- ( ( x = A -> ( ph <-> ps ) ) -> ( ( x = A -> ph ) -> ( x = A -> ps ) ) )
5	4	al2imi 1504	. . . . 5 |- ( A. x ( x = A -> ( ph <-> ps ) ) -> ( A. x ( x = A -> ph ) -> A. x ( x = A -> ps ) ) )
6	5	3ad2ant2 1043	. . . 4 |- ( ( F/ x ps /\ A. x ( x = A -> ( ph <-> ps ) ) /\ A e. V ) -> ( A. x ( x = A -> ph ) -> A. x ( x = A -> ps ) ) )
7		19.23t 1723	. . . . 5 |- ( F/ x ps -> ( A. x ( x = A -> ps ) <-> ( E. x x = A -> ps ) ) )
8	7	3ad2ant1 1042	. . . 4 |- ( ( F/ x ps /\ A. x ( x = A -> ( ph <-> ps ) ) /\ A e. V ) -> ( A. x ( x = A -> ps ) <-> ( E. x x = A -> ps ) ) )
9	6, 8	sylibd 149	. . 3 |- ( ( F/ x ps /\ A. x ( x = A -> ( ph <-> ps ) ) /\ A e. V ) -> ( A. x ( x = A -> ph ) -> ( E. x x = A -> ps ) ) )
10	2, 9	mpid 42	. 2 |- ( ( F/ x ps /\ A. x ( x = A -> ( ph <-> ps ) ) /\ A e. V ) -> ( A. x ( x = A -> ph ) -> ps ) )
11		biimpr 130	. . . . . . 7 |- ( ( ph <-> ps ) -> ( ps -> ph ) )
12	11	imim2i 12	. . . . . 6 |- ( ( x = A -> ( ph <-> ps ) ) -> ( x = A -> ( ps -> ph ) ) )
13	12	com23 78	. . . . 5 |- ( ( x = A -> ( ph <-> ps ) ) -> ( ps -> ( x = A -> ph ) ) )
14	13	alimi 1501	. . . 4 |- ( A. x ( x = A -> ( ph <-> ps ) ) -> A. x ( ps -> ( x = A -> ph ) ) )
15	14	3ad2ant2 1043	. . 3 |- ( ( F/ x ps /\ A. x ( x = A -> ( ph <-> ps ) ) /\ A e. V ) -> A. x ( ps -> ( x = A -> ph ) ) )
16		19.21t 1628	. . . 4 |- ( F/ x ps -> ( A. x ( ps -> ( x = A -> ph ) ) <-> ( ps -> A. x ( x = A -> ph ) ) ) )
17	16	3ad2ant1 1042	. . 3 |- ( ( F/ x ps /\ 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 147	. 2 |- ( ( F/ x ps /\ A. x ( x = A -> ( ph <-> ps ) ) /\ A e. V ) -> ( ps -> A. x ( x = A -> ph ) ) )
19	10, 18	impbid 129	1 |- ( ( F/ x ps /\ A. x ( x = A -> ( ph <-> ps ) ) /\ A e. V ) -> ( A. x ( x = A -> ph ) <-> ps ) )

Syntax hints:   -> wi 4   <-> wb 105   /\ w3a 1002   A.wal 1393   = wceq 1395   F/wnf 1506   E.wex 1538   e. wcel 2200

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1493  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-v 2801

This theorem is referenced by:  ceqsralt  2827