Theorem spesbc 3521

Description: Existence form of spsbc 3448. (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 3445	. . 3 |- ( [. A / x ]. ph -> A e. _V )
2		rspesbca 3520	. . 3 |- ( ( A e. _V /\ [. A / x ]. ph ) -> E. x e. _V ph )
3	1, 2	mpancom 703	. 2 |- ( [. A / x ]. ph -> E. x e. _V ph )
4		rexv 3220	. 2 |- ( E. x e. _V ph <-> E. x ph )
5	3, 4	sylib 208	1 |- ( [. A / x ]. ph -> E. x ph )

Syntax hints:   -> wi 4   E.wex 1704   e. wcel 1990   E.wrex 2913   _Vcvv 3200   [.wsbc 3435

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-v 3202  df-sbc 3436

This theorem is referenced by:  spesbcd  3522  opelopabsb  4985  sbccomieg  37357  frege124d  38053  sbiota1  38635