Theorem ceqsalt 3038

Description: Closed theorem version of ceqsalg 3040. (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 3025	. . . 4 |- ( A e. V -> E. x x = A )
2	1	3ad2ant3 1032	. . 3 |- ( ( F/ x ps /\ A. x ( x = A -> ( ph <-> ps ) ) /\ A e. V ) -> E. x x = A )
3		biimp 198	. . . . . . 7 |- ( ( ph <-> ps ) -> ( ph -> ps ) )
4	3	imim3i 61	. . . . . 6 |- ( ( x = A -> ( ph <-> ps ) ) -> ( ( x = A -> ph ) -> ( x = A -> ps ) ) )
5	4	al2imi 1691	. . . . 5 |- ( A. x ( x = A -> ( ph <-> ps ) ) -> ( A. x ( x = A -> ph ) -> A. x ( x = A -> ps ) ) )
6	5	3ad2ant2 1031	. . . 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 1996	. . . . 5 |- ( F/ x ps -> ( A. x ( x = A -> ps ) <-> ( E. x x = A -> ps ) ) )
8	7	3ad2ant1 1030	. . . 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 222	. . 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 203	. . . . . . 7 |- ( ( ph <-> ps ) -> ( ps -> ph ) )
12	11	imim2i 16	. . . . . 6 |- ( ( x = A -> ( ph <-> ps ) ) -> ( x = A -> ( ps -> ph ) ) )
13	12	com23 81	. . . . 5 |- ( ( x = A -> ( ph <-> ps ) ) -> ( ps -> ( x = A -> ph ) ) )
14	13	alimi 1688	. . . 4 |- ( A. x ( x = A -> ( ph <-> ps ) ) -> A. x ( ps -> ( x = A -> ph ) ) )
15	14	3ad2ant2 1031	. . 3 |- ( ( F/ x ps /\ A. x ( x = A -> ( ph <-> ps ) ) /\ A e. V ) -> A. x ( ps -> ( x = A -> ph ) ) )
16		19.21t 1991	. . . 4 |- ( F/ x ps -> ( A. x ( ps -> ( x = A -> ph ) ) <-> ( ps -> A. x ( x = A -> ph ) ) ) )
17	16	3ad2ant1 1030	. . 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 215	. 2 |- ( ( F/ x ps /\ A. x ( x = A -> ( ph <-> ps ) ) /\ A e. V ) -> ( ps -> A. x ( x = A -> ph ) ) )
19	10, 18	impbid 195	1 |- ( ( F/ x ps /\ A. x ( x = A -> ( ph <-> ps ) ) /\ A e. V ) -> ( A. x ( x = A -> ph ) <-> ps ) )

Syntax hints:   -> wi 4   <-> wb 189   /\ w3a 986   A.wal 1446   = wceq 1448   E.wex 1667   F/wnf 1671   e. wcel 1891

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1673  ax-4 1686  ax-5 1762  ax-6 1809  ax-7 1855  ax-10 1919  ax-12 1937  ax-ext 2432

This theorem depends on definitions:  df-bi 190  df-an 377  df-3an 988  df-tru 1451  df-ex 1668  df-nf 1672  df-sb 1802  df-clab 2439  df-cleq 2445  df-clel 2448  df-v 3015

This theorem is referenced by:  ceqsralt  3039  ceqsalg  3040