Theorem ceqsalt 2756

Description: Closed theorem version of ceqsalg 2758. (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 2744	. . . 4 |- ( A e. V -> E. x x = A )
2	1	3ad2ant3 1015	. . 3 |- ( ( F/ x ps /\ A. x ( x = A -> ( ph <-> ps ) ) /\ A e. V ) -> E. x x = A )
3		biimp 117	. . . . . . 7 |- ( ( ph <-> ps ) -> ( ph -> ps ) )
4	3	imim3i 61	. . . . . 6 |- ( ( x = A -> ( ph <-> ps ) ) -> ( ( x = A -> ph ) -> ( x = A -> ps ) ) )
5	4	al2imi 1451	. . . . 5 |- ( A. x ( x = A -> ( ph <-> ps ) ) -> ( A. x ( x = A -> ph ) -> A. x ( x = A -> ps ) ) )
6	5	3ad2ant2 1014	. . . 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 1670	. . . . 5 |- ( F/ x ps -> ( A. x ( x = A -> ps ) <-> ( E. x x = A -> ps ) ) )
8	7	3ad2ant1 1013	. . . 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 148	. . 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 129	. . . . . . 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 1448	. . . 4 |- ( A. x ( x = A -> ( ph <-> ps ) ) -> A. x ( ps -> ( x = A -> ph ) ) )
15	14	3ad2ant2 1014	. . 3 |- ( ( F/ x ps /\ A. x ( x = A -> ( ph <-> ps ) ) /\ A e. V ) -> A. x ( ps -> ( x = A -> ph ) ) )
16		19.21t 1575	. . . 4 |- ( F/ x ps -> ( A. x ( ps -> ( x = A -> ph ) ) <-> ( ps -> A. x ( x = A -> ph ) ) ) )
17	16	3ad2ant1 1013	. . 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 146	. 2 |- ( ( F/ x ps /\ A. x ( x = A -> ( ph <-> ps ) ) /\ A e. V ) -> ( ps -> A. x ( x = A -> ph ) ) )
19	10, 18	impbid 128	1 |- ( ( F/ x ps /\ A. x ( x = A -> ( ph <-> ps ) ) /\ A e. V ) -> ( A. x ( x = A -> ph ) <-> ps ) )

Syntax hints:   -> wi 4   <-> wb 104   /\ w3a 973   A.wal 1346   = wceq 1348   F/wnf 1453   E.wex 1485   e. wcel 2141

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 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-3an 975  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-v 2732

This theorem is referenced by:  ceqsralt  2757