Theorem spesbc 3040

Description: Existence form of spsbc 2966. (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 2963	. . 3 |- ( [. A / x ]. ph -> A e. _V )
2		rspesbca 3039	. . 3 |- ( ( A e. _V /\ [. A / x ]. ph ) -> E. x e. _V ph )
3	1, 2	mpancom 420	. 2 |- ( [. A / x ]. ph -> E. x e. _V ph )
4		rexv 2748	. 2 |- ( E. x e. _V ph <-> E. x ph )
5	3, 4	sylib 121	1 |- ( [. A / x ]. ph -> E. x ph )

Syntax hints:   -> wi 4   E.wex 1485   e. wcel 2141   E.wrex 2449   _Vcvv 2730   [.wsbc 2955

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rex 2454  df-v 2732  df-sbc 2956

This theorem is referenced by:  spesbcd  3041  opelopabsb  4245