Theorem mob 3388

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 3212	. . . . 5 |- ( B e. D -> B e. _V )
2		nfv 1843	. . . . . . . . . 10 |- F/ x B e. _V
3		nfmo1 2481	. . . . . . . . . 10 |- F/ x E* x ph
4		nfv 1843	. . . . . . . . . 10 |- F/ x ps
5	2, 3, 4	nf3an 1831	. . . . . . . . 9 |- F/ x ( B e. _V /\ E* x ph /\ ps )
6		nfv 1843	. . . . . . . . 9 |- F/ x ( A = B <-> ch )
7	5, 6	nfim 1825	. . . . . . . 8 |- F/ x ( ( B e. _V /\ E* x ph /\ ps ) -> ( A = B <-> ch ) )
8		moi.1	. . . . . . . . . 10 |- ( x = A -> ( ph <-> ps ) )
9	8	3anbi3d 1405	. . . . . . . . 9 |- ( x = A -> ( ( B e. _V /\ E* x ph /\ ph ) <-> ( B e. _V /\ E* x ph /\ ps ) ) )
10		eqeq1 2626	. . . . . . . . . 10 |- ( x = A -> ( x = B <-> A = B ) )
11	10	bibi1d 333	. . . . . . . . 9 |- ( x = A -> ( ( x = B <-> ch ) <-> ( A = B <-> ch ) ) )
12	9, 11	imbi12d 334	. . . . . . . 8 |- ( x = A -> ( ( ( B e. _V /\ E* x ph /\ ph ) -> ( x = B <-> ch ) ) <-> ( ( B e. _V /\ E* x ph /\ ps ) -> ( A = B <-> ch ) ) ) )
13		moi.2	. . . . . . . . 9 |- ( x = B -> ( ph <-> ch ) )
14	13	mob2 3386	. . . . . . . 8 |- ( ( B e. _V /\ E* x ph /\ ph ) -> ( x = B <-> ch ) )
15	7, 12, 14	vtoclg1f 3265	. . . . . . 7 |- ( A e. C -> ( ( B e. _V /\ E* x ph /\ ps ) -> ( A = B <-> ch ) ) )
16	15	com12 32	. . . . . 6 |- ( ( B e. _V /\ E* x ph /\ ps ) -> ( A e. C -> ( A = B <-> ch ) ) )
17	16	3expib 1268	. . . . 5 |- ( B e. _V -> ( ( E* x ph /\ ps ) -> ( A e. C -> ( A = B <-> ch ) ) ) )
18	1, 17	syl 17	. . . 4 |- ( B e. D -> ( ( E* x ph /\ ps ) -> ( A e. C -> ( A = B <-> ch ) ) ) )
19	18	com3r 87	. . 3 |- ( A e. C -> ( B e. D -> ( ( E* x ph /\ ps ) -> ( A = B <-> ch ) ) ) )
20	19	imp 445	. 2 |- ( ( A e. C /\ B e. D ) -> ( ( E* x ph /\ ps ) -> ( A = B <-> ch ) ) )
21	20	3impib 1262	1 |- ( ( ( A e. C /\ B e. D ) /\ E* x ph /\ ps ) -> ( A = B <-> ch ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   E*wmo 2471   _Vcvv 3200

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202

This theorem is referenced by:  moi  3389  rmob  3529