Theorem rmob 3135

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 2623	. 2 |- (E*x e. A ph <-> E*x(x e. A /\ ph))
2		simprl 732	. . . 4 |- ((E*x(x e. A /\ ph) /\ (B e. A /\ ps)) -> B e. A)
3		eleq1 2413	. . . 4 |- (B = C -> (B e. A <-> C e. A))
4	2, 3	syl5ibcom 211	. . 3 |- ((E*x(x e. A /\ ph) /\ (B e. A /\ ps)) -> (B = C -> C e. A))
5		simpl 443	. . . 4 |- ((C e. A /\ ch) -> C e. A)
6	5	a1i 10	. . 3 |- ((E*x(x e. A /\ ph) /\ (B e. A /\ ps)) -> ((C e. A /\ ch) -> C e. A))
7		simplrl 736	. . . . 5 |- (((E*x(x e. A /\ ph) /\ (B e. A /\ ps)) /\ C e. A) -> B e. A)
8		simpr 447	. . . . 5 |- (((E*x(x e. A /\ ph) /\ (B e. A /\ ps)) /\ C e. A) -> C e. A)
9		simpll 730	. . . . 5 |- (((E*x(x e. A /\ ph) /\ (B e. A /\ ps)) /\ C e. A) -> E*x(x e. A /\ ph))
10		simplrr 737	. . . . 5 |- (((E*x(x e. A /\ ph) /\ (B e. A /\ ps)) /\ C e. A) -> ps)
11		eleq1 2413	. . . . . . 7 |- (x = B -> (x e. A <-> B e. A))
12		rmoi.b	. . . . . . 7 |- (x = B -> (ph <-> ps))
13	11, 12	anbi12d 691	. . . . . 6 |- (x = B -> ((x e. A /\ ph) <-> (B e. A /\ ps)))
14		eleq1 2413	. . . . . . 7 |- (x = C -> (x e. A <-> C e. A))
15		rmoi.c	. . . . . . 7 |- (x = C -> (ph <-> ch))
16	14, 15	anbi12d 691	. . . . . 6 |- (x = C -> ((x e. A /\ ph) <-> (C e. A /\ ch)))
17	13, 16	mob 3019	. . . . 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 1192	. . . 4 |- (((E*x(x e. A /\ ph) /\ (B e. A /\ ps)) /\ C e. A) -> (B = C <-> (C e. A /\ ch)))
19	18	ex 423	. . 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 343	. 2 |- ((E*x(x e. A /\ ph) /\ (B e. A /\ ps)) -> (B = C <-> (C e. A /\ ch)))
21	1, 20	sylanb 458	1 |- ((E*x e. A ph /\ (B e. A /\ ps)) -> (B = C <-> (C e. A /\ ch)))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358   = wceq 1642   e. wcel 1710  E*wmo 2205  E*wrmo 2618

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-rmo 2623  df-v 2862

This theorem is referenced by:  rmoi  3136