Theorem mob2 2910

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 994	. . 3 |- ( ( A e. B /\ E* x ph /\ ph ) -> ph )
2		moi2.1	. . 3 |- ( x = A -> ( ph <-> ps ) )
3	1, 2	syl5ibcom 154	. 2 |- ( ( A e. B /\ E* x ph /\ ph ) -> ( x = A -> ps ) )
4		nfs1v 1932	. . . . . . . 8 |- F/ x [ y / x ] ph
5		sbequ12 1764	. . . . . . . 8 |- ( x = y -> ( ph <-> [ y / x ] ph ) )
6	4, 5	mo4f 2079	. . . . . . 7 |- ( E* x ph <-> A. x A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) )
7		sp 1504	. . . . . . 7 |- ( A. x A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) -> A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) )
8	6, 7	sylbi 120	. . . . . 6 |- ( E* x ph -> A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) )
9		nfv 1521	. . . . . . . . . 10 |- F/ x ps
10	9, 2	sbhypf 2779	. . . . . . . . 9 |- ( y = A -> ( [ y / x ] ph <-> ps ) )
11	10	anbi2d 461	. . . . . . . 8 |- ( y = A -> ( ( ph /\ [ y / x ] ph ) <-> ( ph /\ ps ) ) )
12		eqeq2 2180	. . . . . . . 8 |- ( y = A -> ( x = y <-> x = A ) )
13	11, 12	imbi12d 233	. . . . . . 7 |- ( y = A -> ( ( ( ph /\ [ y / x ] ph ) -> x = y ) <-> ( ( ph /\ ps ) -> x = A ) ) )
14	13	spcgv 2817	. . . . . 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 123	. . . 4 |- ( ( A e. B /\ E* x ph ) -> ( ( ph /\ ps ) -> x = A ) )
17	16	expd 256	. . 3 |- ( ( A e. B /\ E* x ph ) -> ( ph -> ( ps -> x = A ) ) )
18	17	3impia 1195	. 2 |- ( ( A e. B /\ E* x ph /\ ph ) -> ( ps -> x = A ) )
19	3, 18	impbid 128	1 |- ( ( A e. B /\ E* x ph /\ ph ) -> ( x = A <-> ps ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 973   A.wal 1346   = wceq 1348   [wsb 1755   E*wmo 2020   e. wcel 2141

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732

This theorem is referenced by:  moi2  2911  mob  2912