Theorem spesbc 3115

Description: Existence form of spsbc 3040. (Contributed by Mario Carneiro, 18-Nov-2016.)

Assertion

Ref	Expression
spesbc	|- ( [. A / x ]. ph -> E. x ph )

Proof of Theorem spesbc

Step	Hyp	Ref	Expression
1		sbcex 3037	. . 3 |- ( [. A / x ]. ph -> A e. _V )
2		rspesbca 3114	. . 3 |- ( ( A e. _V /\ [. A / x ]. ph ) -> E. x e. _V ph )
3	1, 2	mpancom 422	. 2 |- ( [. A / x ]. ph -> E. x e. _V ph )
4		rexv 2818	. 2 |- ( E. x e. _V ph <-> E. x ph )
5	3, 4	sylib 122	1 |- ( [. A / x ]. ph -> E. x ph )

Syntax hints:   -> wi 4   E.wex 1538   e. wcel 2200   E.wrex 2509   _Vcvv 2799   [.wsbc 3028

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rex 2514  df-v 2801  df-sbc 3029

This theorem is referenced by:  spesbcd  3116  opelopabsb  4348