Theorem spesbc 3128

Description: Existence form of spsbc 3059. (Contributed by Mario Carneiro, 18-Nov-2016.)

Assertion

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

Proof of Theorem spesbc

Step	Hyp	Ref	Expression
1		sbcex 3056	. . 3 |- ( [.A / x ].ph -> A e. _V)
2		rspesbca 3127	. . 3 |- ((A e. _V /\ [.A / x ].ph) -> E.x e. _V ph)
3	1, 2	mpancom 650	. 2 |- ( [.A / x ].ph -> E.x e. _V ph)
4		rexv 2874	. 2 |- (E.x e. _V ph <-> E.xph)
5	3, 4	sylib 188	1 |- ( [.A / x ].ph -> E.xph)

Syntax hints:   -> wi 4  E.wex 1541   e. wcel 1710  E.wrex 2616  _Vcvv 2860   [.wsbc 3047

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ral 2620  df-rex 2621  df-v 2862  df-sbc 3048

This theorem is referenced by:  spesbcd  3129  opelopabsb  4698