Theorem eqeu 2821

Description: A condition which implies existential uniqueness. (Contributed by Jeff Hankins, 8-Sep-2009.)

Hypothesis

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

Assertion

Ref	Expression
eqeu	|- ( ( A e. B /\ ps /\ A. x ( ph -> x = A ) ) -> E! x ph )

Distinct variable groups:   ps, x   x, A

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

Proof of Theorem eqeu

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqeu.1	. . . . 5 |- ( x = A -> ( ph <-> ps ) )
2	1	spcegv 2743	. . . 4 |- ( A e. B -> ( ps -> E. x ph ) )
3	2	imp 123	. . 3 |- ( ( A e. B /\ ps ) -> E. x ph )
4	3	3adant3 982	. 2 |- ( ( A e. B /\ ps /\ A. x ( ph -> x = A ) ) -> E. x ph )
5		eqeq2 2122	. . . . . . 7 |- ( y = A -> ( x = y <-> x = A ) )
6	5	imbi2d 229	. . . . . 6 |- ( y = A -> ( ( ph -> x = y ) <-> ( ph -> x = A ) ) )
7	6	albidv 1776	. . . . 5 |- ( y = A -> ( A. x ( ph -> x = y ) <-> A. x ( ph -> x = A ) ) )
8	7	spcegv 2743	. . . 4 |- ( A e. B -> ( A. x ( ph -> x = A ) -> E. y A. x ( ph -> x = y ) ) )
9	8	imp 123	. . 3 |- ( ( A e. B /\ A. x ( ph -> x = A ) ) -> E. y A. x ( ph -> x = y ) )
10	9	3adant2 981	. 2 |- ( ( A e. B /\ ps /\ A. x ( ph -> x = A ) ) -> E. y A. x ( ph -> x = y ) )
11		nfv 1489	. . 3 |- F/ y ph
12	11	eu3 2019	. 2 |- ( E! x ph <-> ( E. x ph /\ E. y A. x ( ph -> x = y ) ) )
13	4, 10, 12	sylanbrc 411	1 |- ( ( A e. B /\ ps /\ A. x ( ph -> x = A ) ) -> E! x ph )

Syntax hints:   -> wi 4   <-> wb 104   /\ w3a 943   A.wal 1310   = wceq 1312   E.wex 1449   e. wcel 1461   E!weu 1973

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095

This theorem depends on definitions:  df-bi 116  df-3an 945  df-tru 1315  df-nf 1418  df-sb 1717  df-eu 1976  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-v 2657

This theorem is referenced by: (None)