Theorem spesbc 3075

Description: Existence form of spsbc 3001. (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 2998	. . 3 |- ( [. A / x ]. ph -> A e. _V )
2		rspesbca 3074	. . 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 2781	. 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 1506   e. wcel 2167   E.wrex 2476   _Vcvv 2763   [.wsbc 2989

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-rex 2481  df-v 2765  df-sbc 2990

This theorem is referenced by:  spesbcd  3076  opelopabsb  4294