Theorem mob 3019

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*xph /\ 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 2868	. . . . 5 |- (B e. D -> B e. _V)
2		nfcv 2490	. . . . . . . 8 |- F/_xA
3		nfv 1619	. . . . . . . . . 10 |- F/x B e. _V
4		nfmo1 2215	. . . . . . . . . 10 |- F/xE*xph
5		nfv 1619	. . . . . . . . . 10 |- F/xps
6	3, 4, 5	nf3an 1827	. . . . . . . . 9 |- F/x(B e. _V /\ E*xph /\ ps)
7		nfv 1619	. . . . . . . . 9 |- F/x(A = B <-> ch)
8	6, 7	nfim 1813	. . . . . . . 8 |- F/x((B e. _V /\ E*xph /\ ps) -> (A = B <-> ch))
9		moi.1	. . . . . . . . . 10 |- (x = A -> (ph <-> ps))
10	9	3anbi3d 1258	. . . . . . . . 9 |- (x = A -> ((B e. _V /\ E*xph /\ ph) <-> (B e. _V /\ E*xph /\ ps)))
11		eqeq1 2359	. . . . . . . . . 10 |- (x = A -> (x = B <-> A = B))
12	11	bibi1d 310	. . . . . . . . 9 |- (x = A -> ((x = B <-> ch) <-> (A = B <-> ch)))
13	10, 12	imbi12d 311	. . . . . . . 8 |- (x = A -> (((B e. _V /\ E*xph /\ ph) -> (x = B <-> ch)) <-> ((B e. _V /\ E*xph /\ ps) -> (A = B <-> ch))))
14		moi.2	. . . . . . . . 9 |- (x = B -> (ph <-> ch))
15	14	mob2 3017	. . . . . . . 8 |- ((B e. _V /\ E*xph /\ ph) -> (x = B <-> ch))
16	2, 8, 13, 15	vtoclgf 2914	. . . . . . 7 |- (A e. C -> ((B e. _V /\ E*xph /\ ps) -> (A = B <-> ch)))
17	16	com12 27	. . . . . 6 |- ((B e. _V /\ E*xph /\ ps) -> (A e. C -> (A = B <-> ch)))
18	17	3expib 1154	. . . . 5 |- (B e. _V -> ((E*xph /\ ps) -> (A e. C -> (A = B <-> ch))))
19	1, 18	syl 15	. . . 4 |- (B e. D -> ((E*xph /\ ps) -> (A e. C -> (A = B <-> ch))))
20	19	com3r 73	. . 3 |- (A e. C -> (B e. D -> ((E*xph /\ ps) -> (A = B <-> ch))))
21	20	imp 418	. 2 |- ((A e. C /\ B e. D) -> ((E*xph /\ ps) -> (A = B <-> ch)))
22	21	3impib 1149	1 |- (((A e. C /\ B e. D) /\ E*xph /\ ps) -> (A = B <-> ch))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358   /\ w3a 934   = wceq 1642   e. wcel 1710  E*wmo 2205  _Vcvv 2860

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862

This theorem is referenced by:  moi  3020  rmob  3135