Theorem eqreu 2784

Description: A condition which implies existential uniqueness. (Contributed by Mario Carneiro, 2-Oct-2015.)

Hypothesis

Ref	Expression
eqreu.1	|- ( x = B -> ( ph <-> ps ) )

Assertion

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

Distinct variable groups:   x, A   x, B   ps, x

Allowed substitution hint:   ph( x)

Proof of Theorem eqreu

Step	Hyp	Ref	Expression
1		ralbiim 2491	. . . . 5 |- ( A. x e. A ( ph <-> x = B ) <-> ( A. x e. A ( ph -> x = B ) /\ A. x e. A ( x = B -> ph ) ) )
2		eqreu.1	. . . . . . 7 |- ( x = B -> ( ph <-> ps ) )
3	2	ceqsralv 2630	. . . . . 6 |- ( B e. A -> ( A. x e. A ( x = B -> ph ) <-> ps ) )
4	3	anbi2d 451	. . . . 5 |- ( B e. A -> ( ( A. x e. A ( ph -> x = B ) /\ A. x e. A ( x = B -> ph ) ) <-> ( A. x e. A ( ph -> x = B ) /\ ps ) ) )
5	1, 4	syl5bb 190	. . . 4 |- ( B e. A -> ( A. x e. A ( ph <-> x = B ) <-> ( A. x e. A ( ph -> x = B ) /\ ps ) ) )
6		reu6i 2783	. . . . 5 |- ( ( B e. A /\ A. x e. A ( ph <-> x = B ) ) -> E! x e. A ph )
7	6	ex 113	. . . 4 |- ( B e. A -> ( A. x e. A ( ph <-> x = B ) -> E! x e. A ph ) )
8	5, 7	sylbird 168	. . 3 |- ( B e. A -> ( ( A. x e. A ( ph -> x = B ) /\ ps ) -> E! x e. A ph ) )
9	8	3impib 1136	. 2 |- ( ( B e. A /\ A. x e. A ( ph -> x = B ) /\ ps ) -> E! x e. A ph )
10	9	3com23 1144	1 |- ( ( B e. A /\ ps /\ A. x e. A ( ph -> x = B ) ) -> E! x e. A ph )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   /\ w3a 919   = wceq 1284   e. wcel 1433   A.wral 2348   E!wreu 2350

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-reu 2355  df-v 2603

This theorem is referenced by: (None)