Theorem eqeu 2996

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 2913	. . . 4 |- ( A e. B -> ( ps -> E. x ph ) )
3	2	imp 124	. . 3 |- ( ( A e. B /\ ps ) -> E. x ph )
4	3	3adant3 1048	. 2 |- ( ( A e. B /\ ps /\ A. x ( ph -> x = A ) ) -> E. x ph )
5		eqeq2 2248	. . . . . . 7 |- ( y = A -> ( x = y <-> x = A ) )
6	5	imbi2d 230	. . . . . 6 |- ( y = A -> ( ( ph -> x = y ) <-> ( ph -> x = A ) ) )
7	6	albidv 1877	. . . . 5 |- ( y = A -> ( A. x ( ph -> x = y ) <-> A. x ( ph -> x = A ) ) )
8	7	spcegv 2913	. . . 4 |- ( A e. B -> ( A. x ( ph -> x = A ) -> E. y A. x ( ph -> x = y ) ) )
9	8	imp 124	. . 3 |- ( ( A e. B /\ A. x ( ph -> x = A ) ) -> E. y A. x ( ph -> x = y ) )
10	9	3adant2 1047	. 2 |- ( ( A e. B /\ ps /\ A. x ( ph -> x = A ) ) -> E. y A. x ( ph -> x = y ) )
11		nfv 1581	. . 3 |- F/ y ph
12	11	eu3 2133	. 2 |- ( E! x ph <-> ( E. x ph /\ E. y A. x ( ph -> x = y ) ) )
13	4, 10, 12	sylanbrc 421	1 |- ( ( A e. B /\ ps /\ A. x ( ph -> x = A ) ) -> E! x ph )

Syntax hints:   -> wi 4   <-> wb 105   /\ w3a 1009   A.wal 1400   = wceq 1402   E.wex 1545   E!weu 2086   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-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823

This theorem is referenced by: (None)