Theorem mob 2866

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 2697	. . . . 5 |- ( B e. D -> B e. _V )
2		nfcv 2281	. . . . . . . 8 |- F/_ x A
3		nfv 1508	. . . . . . . . . 10 |- F/ x B e. _V
4		nfmo1 2011	. . . . . . . . . 10 |- F/ x E* x ph
5		nfv 1508	. . . . . . . . . 10 |- F/ x ps
6	3, 4, 5	nf3an 1545	. . . . . . . . 9 |- F/ x ( B e. _V /\ E* x ph /\ ps )
7		nfv 1508	. . . . . . . . 9 |- F/ x ( A = B <-> ch )
8	6, 7	nfim 1551	. . . . . . . 8 |- F/ x ( ( B e. _V /\ E* x ph /\ ps ) -> ( A = B <-> ch ) )
9		moi.1	. . . . . . . . . 10 |- ( x = A -> ( ph <-> ps ) )
10	9	3anbi3d 1296	. . . . . . . . 9 |- ( x = A -> ( ( B e. _V /\ E* x ph /\ ph ) <-> ( B e. _V /\ E* x ph /\ ps ) ) )
11		eqeq1 2146	. . . . . . . . . 10 |- ( x = A -> ( x = B <-> A = B ) )
12	11	bibi1d 232	. . . . . . . . 9 |- ( x = A -> ( ( x = B <-> ch ) <-> ( A = B <-> ch ) ) )
13	10, 12	imbi12d 233	. . . . . . . 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 2864	. . . . . . . 8 |- ( ( B e. _V /\ E* x ph /\ ph ) -> ( x = B <-> ch ) )
16	2, 8, 13, 15	vtoclgf 2744	. . . . . . 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 1184	. . . . 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 123	. 2 |- ( ( A e. C /\ B e. D ) -> ( ( E* x ph /\ ps ) -> ( A = B <-> ch ) ) )
22	21	3impib 1179	1 |- ( ( ( A e. C /\ B e. D ) /\ E* x ph /\ ps ) -> ( A = B <-> ch ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 962   = wceq 1331   e. wcel 1480   E*wmo 2000   _Vcvv 2686

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-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688

This theorem is referenced by:  moi  2867  rmob  3001