Theorem mob2 3386

Description: Consequence of "at most one." (Contributed by NM, 2-Jan-2015.)

Hypothesis

Ref	Expression
moi2.1	|- ( x = A -> ( ph <-> ps ) )

Assertion

Ref	Expression
mob2	|- ( ( A e. B /\ E* x ph /\ ph ) -> ( x = A <-> ps ) )

Distinct variable groups:   x, A   ps, x

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

Proof of Theorem mob2

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp3 1063	. . 3 |- ( ( A e. B /\ E* x ph /\ ph ) -> ph )
2		moi2.1	. . 3 |- ( x = A -> ( ph <-> ps ) )
3	1, 2	syl5ibcom 235	. 2 |- ( ( A e. B /\ E* x ph /\ ph ) -> ( x = A -> ps ) )
4		nfv 1843	. . . . . . . . 9 |- F/ x ps
5	4, 2	sbhypf 3253	. . . . . . . 8 |- ( y = A -> ( [ y / x ] ph <-> ps ) )
6	5	anbi2d 740	. . . . . . 7 |- ( y = A -> ( ( ph /\ [ y / x ] ph ) <-> ( ph /\ ps ) ) )
7		eqeq2 2633	. . . . . . 7 |- ( y = A -> ( x = y <-> x = A ) )
8	6, 7	imbi12d 334	. . . . . 6 |- ( y = A -> ( ( ( ph /\ [ y / x ] ph ) -> x = y ) <-> ( ( ph /\ ps ) -> x = A ) ) )
9	8	spcgv 3293	. . . . 5 |- ( A e. B -> ( A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) -> ( ( ph /\ ps ) -> x = A ) ) )
10		nfs1v 2437	. . . . . . 7 |- F/ x [ y / x ] ph
11		sbequ12 2111	. . . . . . 7 |- ( x = y -> ( ph <-> [ y / x ] ph ) )
12	10, 11	mo4f 2516	. . . . . 6 |- ( E* x ph <-> A. x A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) )
13		sp 2053	. . . . . 6 |- ( A. x A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) -> A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) )
14	12, 13	sylbi 207	. . . . 5 |- ( E* x ph -> A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) )
15	9, 14	impel 485	. . . 4 |- ( ( A e. B /\ E* x ph ) -> ( ( ph /\ ps ) -> x = A ) )
16	15	expd 452	. . 3 |- ( ( A e. B /\ E* x ph ) -> ( ph -> ( ps -> x = A ) ) )
17	16	3impia 1261	. 2 |- ( ( A e. B /\ E* x ph /\ ph ) -> ( ps -> x = A ) )
18	3, 17	impbid 202	1 |- ( ( A e. B /\ E* x ph /\ ph ) -> ( x = A <-> ps ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   A.wal 1481   = wceq 1483   [wsb 1880   e. wcel 1990   E*wmo 2471

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-v 3202

This theorem is referenced by:  moi2  3387  mob  3388  rmob2  3531