Theorem rmob 3005

Description: Consequence of "at most one", using implicit substitution. (Contributed by NM, 2-Jan-2015.) (Revised by NM, 16-Jun-2017.)

Hypotheses

Ref	Expression
rmoi.b	|- ( x = B -> ( ph <-> ps ) )
rmoi.c	|- ( x = C -> ( ph <-> ch ) )

Assertion

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

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

Allowed substitution hint:   ph( x)

Proof of Theorem rmob

Step	Hyp	Ref	Expression
1		df-rmo 2425	. 2 |- ( E* x e. A ph <-> E* x ( x e. A /\ ph ) )
2		simprl 521	. . . 4 |- ( ( E* x ( x e. A /\ ph ) /\ ( B e. A /\ ps ) ) -> B e. A )
3		eleq1 2203	. . . 4 |- ( B = C -> ( B e. A <-> C e. A ) )
4	2, 3	syl5ibcom 154	. . 3 |- ( ( E* x ( x e. A /\ ph ) /\ ( B e. A /\ ps ) ) -> ( B = C -> C e. A ) )
5		simpl 108	. . . 4 |- ( ( C e. A /\ ch ) -> C e. A )
6	5	a1i 9	. . 3 |- ( ( E* x ( x e. A /\ ph ) /\ ( B e. A /\ ps ) ) -> ( ( C e. A /\ ch ) -> C e. A ) )
7		simplrl 525	. . . . 5 |- ( ( ( E* x ( x e. A /\ ph ) /\ ( B e. A /\ ps ) ) /\ C e. A ) -> B e. A )
8		simpr 109	. . . . 5 |- ( ( ( E* x ( x e. A /\ ph ) /\ ( B e. A /\ ps ) ) /\ C e. A ) -> C e. A )
9		simpll 519	. . . . 5 |- ( ( ( E* x ( x e. A /\ ph ) /\ ( B e. A /\ ps ) ) /\ C e. A ) -> E* x ( x e. A /\ ph ) )
10		simplrr 526	. . . . 5 |- ( ( ( E* x ( x e. A /\ ph ) /\ ( B e. A /\ ps ) ) /\ C e. A ) -> ps )
11		eleq1 2203	. . . . . . 7 |- ( x = B -> ( x e. A <-> B e. A ) )
12		rmoi.b	. . . . . . 7 |- ( x = B -> ( ph <-> ps ) )
13	11, 12	anbi12d 465	. . . . . 6 |- ( x = B -> ( ( x e. A /\ ph ) <-> ( B e. A /\ ps ) ) )
14		eleq1 2203	. . . . . . 7 |- ( x = C -> ( x e. A <-> C e. A ) )
15		rmoi.c	. . . . . . 7 |- ( x = C -> ( ph <-> ch ) )
16	14, 15	anbi12d 465	. . . . . 6 |- ( x = C -> ( ( x e. A /\ ph ) <-> ( C e. A /\ ch ) ) )
17	13, 16	mob 2870	. . . . 5 |- ( ( ( B e. A /\ C e. A ) /\ E* x ( x e. A /\ ph ) /\ ( B e. A /\ ps ) ) -> ( B = C <-> ( C e. A /\ ch ) ) )
18	7, 8, 9, 7, 10, 17	syl212anc 1227	. . . 4 |- ( ( ( E* x ( x e. A /\ ph ) /\ ( B e. A /\ ps ) ) /\ C e. A ) -> ( B = C <-> ( C e. A /\ ch ) ) )
19	18	ex 114	. . 3 |- ( ( E* x ( x e. A /\ ph ) /\ ( B e. A /\ ps ) ) -> ( C e. A -> ( B = C <-> ( C e. A /\ ch ) ) ) )
20	4, 6, 19	pm5.21ndd 695	. 2 |- ( ( E* x ( x e. A /\ ph ) /\ ( B e. A /\ ps ) ) -> ( B = C <-> ( C e. A /\ ch ) ) )
21	1, 20	sylanb 282	1 |- ( ( E* x e. A ph /\ ( B e. A /\ ps ) ) -> ( B = C <-> ( C e. A /\ ch ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1332   e. wcel 1481   E*wmo 2001   E*wrmo 2420

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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-rmo 2425  df-v 2691

This theorem is referenced by:  rmoi  3006