Theorem rmob2 3531

Description: Consequence of "restricted at most one." (Contributed by Thierry Arnoux, 9-Dec-2019.)

Hypotheses

Ref	Expression
rmoi2.1	|- ( x = B -> ( ps <-> ch ) )
rmoi2.2	|- ( ph -> B e. A )
rmoi2.3	|- ( ph -> E* x e. A ps )
rmoi2.4	|- ( ph -> x e. A )
rmoi2.5	|- ( ph -> ps )

Assertion

Ref	Expression
rmob2	|- ( ph -> ( x = B <-> ch ) )

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

Allowed substitution hints:   ph( x)   ps( x)

Proof of Theorem rmob2

Step	Hyp	Ref	Expression
1		rmoi2.2	. 2 |- ( ph -> B e. A )
2		rmoi2.3	. . . 4 |- ( ph -> E* x e. A ps )
3		df-rmo 2920	. . . 4 |- ( E* x e. A ps <-> E* x ( x e. A /\ ps ) )
4	2, 3	sylib 208	. . 3 |- ( ph -> E* x ( x e. A /\ ps ) )
5		rmoi2.4	. . 3 |- ( ph -> x e. A )
6		rmoi2.5	. . 3 |- ( ph -> ps )
7		eleq1 2689	. . . . 5 |- ( x = B -> ( x e. A <-> B e. A ) )
8		rmoi2.1	. . . . 5 |- ( x = B -> ( ps <-> ch ) )
9	7, 8	anbi12d 747	. . . 4 |- ( x = B -> ( ( x e. A /\ ps ) <-> ( B e. A /\ ch ) ) )
10	9	mob2 3386	. . 3 |- ( ( B e. A /\ E* x ( x e. A /\ ps ) /\ ( x e. A /\ ps ) ) -> ( x = B <-> ( B e. A /\ ch ) ) )
11	1, 4, 5, 6, 10	syl112anc 1330	. 2 |- ( ph -> ( x = B <-> ( B e. A /\ ch ) ) )
12	1, 11	mpbirand 530	1 |- ( ph -> ( x = B <-> 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:  rmoi2  3532