Theorem rmob 3529

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 2920	. 2 |- ( E* x e. A ph <-> E* x ( x e. A /\ ph ) )
2		simprl 794	. . . 4 |- ( ( E* x ( x e. A /\ ph ) /\ ( B e. A /\ ps ) ) -> B e. A )
3		eleq1 2689	. . . 4 |- ( B = C -> ( B e. A <-> C e. A ) )
4	2, 3	syl5ibcom 235	. . 3 |- ( ( E* x ( x e. A /\ ph ) /\ ( B e. A /\ ps ) ) -> ( B = C -> C e. A ) )
5		simpl 473	. . . 4 |- ( ( C e. A /\ ch ) -> C e. A )
6	5	a1i 11	. . 3 |- ( ( E* x ( x e. A /\ ph ) /\ ( B e. A /\ ps ) ) -> ( ( C e. A /\ ch ) -> C e. A ) )
7	2	anim1i 592	. . . . 5 |- ( ( ( E* x ( x e. A /\ ph ) /\ ( B e. A /\ ps ) ) /\ C e. A ) -> ( B e. A /\ C e. A ) )
8		simpll 790	. . . . 5 |- ( ( ( E* x ( x e. A /\ ph ) /\ ( B e. A /\ ps ) ) /\ C e. A ) -> E* x ( x e. A /\ ph ) )
9		simplr 792	. . . . 5 |- ( ( ( E* x ( x e. A /\ ph ) /\ ( B e. A /\ ps ) ) /\ C e. A ) -> ( B e. A /\ ps ) )
10		eleq1 2689	. . . . . . 7 |- ( x = B -> ( x e. A <-> B e. A ) )
11		rmoi.b	. . . . . . 7 |- ( x = B -> ( ph <-> ps ) )
12	10, 11	anbi12d 747	. . . . . 6 |- ( x = B -> ( ( x e. A /\ ph ) <-> ( B e. A /\ ps ) ) )
13		eleq1 2689	. . . . . . 7 |- ( x = C -> ( x e. A <-> C e. A ) )
14		rmoi.c	. . . . . . 7 |- ( x = C -> ( ph <-> ch ) )
15	13, 14	anbi12d 747	. . . . . 6 |- ( x = C -> ( ( x e. A /\ ph ) <-> ( C e. A /\ ch ) ) )
16	12, 15	mob 3388	. . . . 5 |- ( ( ( B e. A /\ C e. A ) /\ E* x ( x e. A /\ ph ) /\ ( B e. A /\ ps ) ) -> ( B = C <-> ( C e. A /\ ch ) ) )
17	7, 8, 9, 16	syl3anc 1326	. . . 4 |- ( ( ( E* x ( x e. A /\ ph ) /\ ( B e. A /\ ps ) ) /\ C e. A ) -> ( B = C <-> ( C e. A /\ ch ) ) )
18	17	ex 450	. . 3 |- ( ( E* x ( x e. A /\ ph ) /\ ( B e. A /\ ps ) ) -> ( C e. A -> ( B = C <-> ( C e. A /\ ch ) ) ) )
19	4, 6, 18	pm5.21ndd 369	. 2 |- ( ( E* x ( x e. A /\ ph ) /\ ( B e. A /\ ps ) ) -> ( B = C <-> ( C e. A /\ ch ) ) )
20	1, 19	sylanb 489	1 |- ( ( E* x e. A ph /\ ( B e. A /\ ps ) ) -> ( B = C <-> ( C e. A /\ ch ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   E*wmo 2471   E*wrmo 2915

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-rmo 2920  df-v 3202

This theorem is referenced by:  rmoi  3530