Theorem mob2 2721

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 906	. . 3 |- ( ( A e. B /\ E* x ph /\ ph ) -> ph )
2		moi2.1	. . 3 |- ( x = A -> ( ph <-> ps ) )
3	1, 2	syl5ibcom 144	. 2 |- ( ( A e. B /\ E* x ph /\ ph ) -> ( x = A -> ps ) )
4		nfs1v 1815	. . . . . . . 8 |- F/ x [ y / x ] ph
5		sbequ12 1654	. . . . . . . 8 |- ( x = y -> ( ph <-> [ y / x ] ph ) )
6	4, 5	mo4f 1960	. . . . . . 7 |- ( E* x ph <-> A. x A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) )
7		sp 1401	. . . . . . 7 |- ( A. x A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) -> A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) )
8	6, 7	sylbi 114	. . . . . 6 |- ( E* x ph -> A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) )
9		nfv 1421	. . . . . . . . . 10 |- F/ x ps
10	9, 2	sbhypf 2603	. . . . . . . . 9 |- ( y = A -> ( [ y / x ] ph <-> ps ) )
11	10	anbi2d 437	. . . . . . . 8 |- ( y = A -> ( ( ph /\ [ y / x ] ph ) <-> ( ph /\ ps ) ) )
12		eqeq2 2049	. . . . . . . 8 |- ( y = A -> ( x = y <-> x = A ) )
13	11, 12	imbi12d 223	. . . . . . 7 |- ( y = A -> ( ( ( ph /\ [ y / x ] ph ) -> x = y ) <-> ( ( ph /\ ps ) -> x = A ) ) )
14	13	spcgv 2640	. . . . . 6 |- ( A e. B -> ( A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) -> ( ( ph /\ ps ) -> x = A ) ) )
15	8, 14	syl5 28	. . . . 5 |- ( A e. B -> ( E* x ph -> ( ( ph /\ ps ) -> x = A ) ) )
16	15	imp 115	. . . 4 |- ( ( A e. B /\ E* x ph ) -> ( ( ph /\ ps ) -> x = A ) )
17	16	expd 245	. . 3 |- ( ( A e. B /\ E* x ph ) -> ( ph -> ( ps -> x = A ) ) )
18	17	3impia 1101	. 2 |- ( ( A e. B /\ E* x ph /\ ph ) -> ( ps -> x = A ) )
19	3, 18	impbid 120	1 |- ( ( A e. B /\ E* x ph /\ ph ) -> ( x = A <-> ps ) )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   /\ w3a 885   A.wal 1241   = wceq 1243   e. wcel 1393   [wsb 1645   E*wmo 1901

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559

This theorem is referenced by:  moi2  2722  mob  2723