Theorem mob 2955

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 2783	. . . . 5 |- ( B e. D -> B e. _V )
2		nfcv 2348	. . . . . . . 8 |- F/_ x A
3		nfv 1551	. . . . . . . . . 10 |- F/ x B e. _V
4		nfmo1 2066	. . . . . . . . . 10 |- F/ x E* x ph
5		nfv 1551	. . . . . . . . . 10 |- F/ x ps
6	3, 4, 5	nf3an 1589	. . . . . . . . 9 |- F/ x ( B e. _V /\ E* x ph /\ ps )
7		nfv 1551	. . . . . . . . 9 |- F/ x ( A = B <-> ch )
8	6, 7	nfim 1595	. . . . . . . 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 2212	. . . . . . . . . 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 2953	. . . . . . . 8 |- ( ( B e. _V /\ E* x ph /\ ph ) -> ( x = B <-> ch ) )
16	2, 8, 13, 15	vtoclgf 2831	. . . . . . 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 2055   e. wcel 2176   _Vcvv 2772

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774

This theorem is referenced by:  moi  2956  rmob  3091  2omotaplemst  7372