Theorem mob2 3006

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 1030	. . 3 |- ( ( A e. B /\ E* x ph /\ ph ) -> ph )
2		moi2.1	. . 3 |- ( x = A -> ( ph <-> ps ) )
3	1, 2	syl5ibcom 155	. 2 |- ( ( A e. B /\ E* x ph /\ ph ) -> ( x = A -> ps ) )
4		nfs1v 1999	. . . . . . . 8 |- F/ x [ y / x ] ph
5		sbequ12 1824	. . . . . . . 8 |- ( x = y -> ( ph <-> [ y / x ] ph ) )
6	4, 5	mo4f 2147	. . . . . . 7 |- ( E* x ph <-> A. x A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) )
7		sp 1564	. . . . . . 7 |- ( A. x A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) -> A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) )
8	6, 7	sylbi 121	. . . . . 6 |- ( E* x ph -> A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) )
9		nfv 1581	. . . . . . . . . 10 |- F/ x ps
10	9, 2	sbhypf 2872	. . . . . . . . 9 |- ( y = A -> ( [ y / x ] ph <-> ps ) )
11	10	anbi2d 468	. . . . . . . 8 |- ( y = A -> ( ( ph /\ [ y / x ] ph ) <-> ( ph /\ ps ) ) )
12		eqeq2 2248	. . . . . . . 8 |- ( y = A -> ( x = y <-> x = A ) )
13	11, 12	imbi12d 234	. . . . . . 7 |- ( y = A -> ( ( ( ph /\ [ y / x ] ph ) -> x = y ) <-> ( ( ph /\ ps ) -> x = A ) ) )
14	13	spcgv 2912	. . . . . 6 |- ( A e. B -> ( A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) -> ( ( ph /\ ps ) -> x = A ) ) )
15	8, 14	syl5 32	. . . . 5 |- ( A e. B -> ( E* x ph -> ( ( ph /\ ps ) -> x = A ) ) )
16	15	imp 124	. . . 4 |- ( ( A e. B /\ E* x ph ) -> ( ( ph /\ ps ) -> x = A ) )
17	16	expd 258	. . 3 |- ( ( A e. B /\ E* x ph ) -> ( ph -> ( ps -> x = A ) ) )
18	17	3impia 1231	. 2 |- ( ( A e. B /\ E* x ph /\ ph ) -> ( ps -> x = A ) )
19	3, 18	impbid 129	1 |- ( ( A e. B /\ E* x ph /\ ph ) -> ( x = A <-> ps ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   A.wal 1400   = wceq 1402   [wsb 1815   E*wmo 2087   e. wcel 2209

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823

This theorem is referenced by:  moi2  3007  mob  3008