Theorem eqeu 2900

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 2818	. . . 4 |- ( A e. B -> ( ps -> E. x ph ) )
3	2	imp 123	. . 3 |- ( ( A e. B /\ ps ) -> E. x ph )
4	3	3adant3 1012	. 2 |- ( ( A e. B /\ ps /\ A. x ( ph -> x = A ) ) -> E. x ph )
5		eqeq2 2180	. . . . . . 7 |- ( y = A -> ( x = y <-> x = A ) )
6	5	imbi2d 229	. . . . . 6 |- ( y = A -> ( ( ph -> x = y ) <-> ( ph -> x = A ) ) )
7	6	albidv 1817	. . . . 5 |- ( y = A -> ( A. x ( ph -> x = y ) <-> A. x ( ph -> x = A ) ) )
8	7	spcegv 2818	. . . 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 1011	. 2 |- ( ( A e. B /\ ps /\ A. x ( ph -> x = A ) ) -> E. y A. x ( ph -> x = y ) )
11		nfv 1521	. . 3 |- F/ y ph
12	11	eu3 2065	. 2 |- ( E! x ph <-> ( E. x ph /\ E. y A. x ( ph -> x = y ) ) )
13	4, 10, 12	sylanbrc 415	1 |- ( ( A e. B /\ ps /\ A. x ( ph -> x = A ) ) -> E! x ph )

Syntax hints:   -> wi 4   <-> wb 104   /\ w3a 973   A.wal 1346   = wceq 1348   E.wex 1485   E!weu 2019   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-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732

This theorem is referenced by: (None)