Theorem eqeu 3377

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 3294	. . . 4 |- ( A e. B -> ( ps -> E. x ph ) )
3	2	imp 445	. . 3 |- ( ( A e. B /\ ps ) -> E. x ph )
4	3	3adant3 1081	. 2 |- ( ( A e. B /\ ps /\ A. x ( ph -> x = A ) ) -> E. x ph )
5		eqeq2 2633	. . . . . . 7 |- ( y = A -> ( x = y <-> x = A ) )
6	5	imbi2d 330	. . . . . 6 |- ( y = A -> ( ( ph -> x = y ) <-> ( ph -> x = A ) ) )
7	6	albidv 1849	. . . . 5 |- ( y = A -> ( A. x ( ph -> x = y ) <-> A. x ( ph -> x = A ) ) )
8	7	spcegv 3294	. . . 4 |- ( A e. B -> ( A. x ( ph -> x = A ) -> E. y A. x ( ph -> x = y ) ) )
9	8	imp 445	. . 3 |- ( ( A e. B /\ A. x ( ph -> x = A ) ) -> E. y A. x ( ph -> x = y ) )
10	9	3adant2 1080	. 2 |- ( ( A e. B /\ ps /\ A. x ( ph -> x = A ) ) -> E. y A. x ( ph -> x = y ) )
11		eu3v 2498	. 2 |- ( E! x ph <-> ( E. x ph /\ E. y A. x ( ph -> x = y ) ) )
12	4, 10, 11	sylanbrc 698	1 |- ( ( A e. B /\ ps /\ A. x ( ph -> x = A ) ) -> E! x ph )

Syntax hints:   -> wi 4   <-> wb 196   /\ w3a 1037   A.wal 1481   = wceq 1483   E.wex 1704   e. wcel 1990   E!weu 2470

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202

This theorem is referenced by:  rngurd  29788  neibastop3  32357  upixp  33524  zrinitorngc  42000  zrtermorngc  42001  zrtermoringc  42070