Theorem reu6i 2732

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 2049	. . . . 5 |- ( y = B -> ( x = y <-> x = B ) )
2	1	bibi2d 221	. . . 4 |- ( y = B -> ( ( ph <-> x = y ) <-> ( ph <-> x = B ) ) )
3	2	ralbidv 2326	. . 3 |- ( y = B -> ( A. x e. A ( ph <-> x = y ) <-> A. x e. A ( ph <-> x = B ) ) )
4	3	rspcev 2656	. 2 |- ( ( B e. A /\ A. x e. A ( ph <-> x = B ) ) -> E. y e. A A. x e. A ( ph <-> x = y ) )
5		reu6 2730	. 2 |- ( E! x e. A ph <-> E. y e. A A. x e. A ( ph <-> x = y ) )
6	4, 5	sylibr 137	1 |- ( ( B e. A /\ A. x e. A ( ph <-> x = B ) ) -> E! x e. A ph )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   = wceq 1243   e. wcel 1393   A.wral 2306   E.wrex 2307   E!wreu 2308

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-reu 2313  df-v 2559

This theorem is referenced by:  eqreu  2733  riota5f  5492  negeu  7202  creur  7911  creui  7912