Theorem eqeu 2943

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 2861	. . . 4 |- ( A e. B -> ( ps -> E. x ph ) )
3	2	imp 124	. . 3 |- ( ( A e. B /\ ps ) -> E. x ph )
4	3	3adant3 1020	. 2 |- ( ( A e. B /\ ps /\ A. x ( ph -> x = A ) ) -> E. x ph )
5		eqeq2 2215	. . . . . . 7 |- ( y = A -> ( x = y <-> x = A ) )
6	5	imbi2d 230	. . . . . 6 |- ( y = A -> ( ( ph -> x = y ) <-> ( ph -> x = A ) ) )
7	6	albidv 1847	. . . . 5 |- ( y = A -> ( A. x ( ph -> x = y ) <-> A. x ( ph -> x = A ) ) )
8	7	spcegv 2861	. . . 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 1019	. 2 |- ( ( A e. B /\ ps /\ A. x ( ph -> x = A ) ) -> E. y A. x ( ph -> x = y ) )
11		nfv 1551	. . 3 |- F/ y ph
12	11	eu3 2100	. 2 |- ( E! x ph <-> ( E. x ph /\ E. y A. x ( ph -> x = y ) ) )
13	4, 10, 12	sylanbrc 417	1 |- ( ( A e. B /\ ps /\ A. x ( ph -> x = A ) ) -> E! x ph )

Syntax hints:   -> wi 4   <-> wb 105   /\ w3a 981   A.wal 1371   = wceq 1373   E.wex 1515   E!weu 2054   e. wcel 2176

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774

This theorem is referenced by: (None)