Theorem reu6i 2783

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

Assertion

Ref	Expression
reu6i	|- ( ( B e. A /\ A. x e. A ( ph <-> x = B ) ) -> E! x e. A ph )

Distinct variable groups:   x, A   x, B

Allowed substitution hint:   ph( x)

Proof of Theorem reu6i

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqeq2 2090	. . . . 5 |- ( y = B -> ( x = y <-> x = B ) )
2	1	bibi2d 230	. . . 4 |- ( y = B -> ( ( ph <-> x = y ) <-> ( ph <-> x = B ) ) )
3	2	ralbidv 2368	. . 3 |- ( y = B -> ( A. x e. A ( ph <-> x = y ) <-> A. x e. A ( ph <-> x = B ) ) )
4	3	rspcev 2701	. 2 |- ( ( B e. A /\ A. x e. A ( ph <-> x = B ) ) -> E. y e. A A. x e. A ( ph <-> x = y ) )
5		reu6 2781	. 2 |- ( E! x e. A ph <-> E. y e. A A. x e. A ( ph <-> x = y ) )
6	4, 5	sylibr 132	1 |- ( ( B e. A /\ A. x e. A ( ph <-> x = B ) ) -> E! x e. A ph )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1284   e. wcel 1433   A.wral 2348   E.wrex 2349   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-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:  eqreu  2784  riota5f  5512  negeu  7299  creur  8036  creui  8037