Theorem mob 2962

Description: Equality implied by "at most one". (Contributed by NM, 18-Feb-2006.)

Hypotheses

Ref	Expression
moi.1	|- ( x = A -> ( ph <-> ps ) )
moi.2	|- ( x = B -> ( ph <-> ch ) )

Assertion

Ref	Expression
mob	|- ( ( ( A e. C /\ B e. D ) /\ E* x ph /\ ps ) -> ( A = B <-> ch ) )

Distinct variable groups:   x, A   x, B   ch, x   ps, x

Allowed substitution hints:   ph( x)   C( x)   D( x)

Proof of Theorem mob

Step	Hyp	Ref	Expression
1		elex 2788	. . . . 5 |- ( B e. D -> B e. _V )
2		nfcv 2350	. . . . . . . 8 |- F/_ x A
3		nfv 1552	. . . . . . . . . 10 |- F/ x B e. _V
4		nfmo1 2067	. . . . . . . . . 10 |- F/ x E* x ph
5		nfv 1552	. . . . . . . . . 10 |- F/ x ps
6	3, 4, 5	nf3an 1590	. . . . . . . . 9 |- F/ x ( B e. _V /\ E* x ph /\ ps )
7		nfv 1552	. . . . . . . . 9 |- F/ x ( A = B <-> ch )
8	6, 7	nfim 1596	. . . . . . . 8 |- F/ x ( ( B e. _V /\ E* x ph /\ ps ) -> ( A = B <-> ch ) )
9		moi.1	. . . . . . . . . 10 |- ( x = A -> ( ph <-> ps ) )
10	9	3anbi3d 1331	. . . . . . . . 9 |- ( x = A -> ( ( B e. _V /\ E* x ph /\ ph ) <-> ( B e. _V /\ E* x ph /\ ps ) ) )
11		eqeq1 2214	. . . . . . . . . 10 |- ( x = A -> ( x = B <-> A = B ) )
12	11	bibi1d 233	. . . . . . . . 9 |- ( x = A -> ( ( x = B <-> ch ) <-> ( A = B <-> ch ) ) )
13	10, 12	imbi12d 234	. . . . . . . 8 |- ( x = A -> ( ( ( B e. _V /\ E* x ph /\ ph ) -> ( x = B <-> ch ) ) <-> ( ( B e. _V /\ E* x ph /\ ps ) -> ( A = B <-> ch ) ) ) )
14		moi.2	. . . . . . . . 9 |- ( x = B -> ( ph <-> ch ) )
15	14	mob2 2960	. . . . . . . 8 |- ( ( B e. _V /\ E* x ph /\ ph ) -> ( x = B <-> ch ) )
16	2, 8, 13, 15	vtoclgf 2836	. . . . . . 7 |- ( A e. C -> ( ( B e. _V /\ E* x ph /\ ps ) -> ( A = B <-> ch ) ) )
17	16	com12 30	. . . . . 6 |- ( ( B e. _V /\ E* x ph /\ ps ) -> ( A e. C -> ( A = B <-> ch ) ) )
18	17	3expib 1209	. . . . 5 |- ( B e. _V -> ( ( E* x ph /\ ps ) -> ( A e. C -> ( A = B <-> ch ) ) ) )
19	1, 18	syl 14	. . . 4 |- ( B e. D -> ( ( E* x ph /\ ps ) -> ( A e. C -> ( A = B <-> ch ) ) ) )
20	19	com3r 79	. . 3 |- ( A e. C -> ( B e. D -> ( ( E* x ph /\ ps ) -> ( A = B <-> ch ) ) ) )
21	20	imp 124	. 2 |- ( ( A e. C /\ B e. D ) -> ( ( E* x ph /\ ps ) -> ( A = B <-> ch ) ) )
22	21	3impib 1204	1 |- ( ( ( A e. C /\ B e. D ) /\ E* x ph /\ ps ) -> ( A = B <-> ch ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 981   = wceq 1373   E*wmo 2056   e. wcel 2178   _Vcvv 2776

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-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778

This theorem is referenced by:  moi  2963  rmob  3099  2omotaplemst  7405