Theorem spesbc 3048

Description: Existence form of spsbc 2974. (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 2971	. . 3 |- ( [. A / x ]. ph -> A e. _V )
2		rspesbca 3047	. . 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 2755	. 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 1492   e. wcel 2148   E.wrex 2456   _Vcvv 2737   [.wsbc 2962

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-v 2739  df-sbc 2963

This theorem is referenced by:  spesbcd  3049  opelopabsb  4260