Theorem ra4esbca 1995

Description: Existence form of ra4sbca 1994.

Assertion

Ref	Expression
ra4esbca	|- ((A e. B /\ [A / x]ph) -> E.x e. B ph)

Distinct variable group:   x,B

Proof of Theorem ra4esbca

Step	Hyp	Ref	Expression
1		ra4sbc 1993	. . . . 5 |- (A e. B -> (A.x e. B -. ph -> [A / x] -. ph))
2		sbcng 1965	. . . . 5 |- (A e. B -> ([A / x] -. ph <-> -. [A / x]ph))
3	1, 2	sylibd 202	. . . 4 |- (A e. B -> (A.x e. B -. ph -> -. [A / x]ph))
4		ralnex 1650	. . . 4 |- (A.x e. B -. ph <-> -. E.x e. B ph)
5	3, 4	syl5ibr 207	. . 3 |- (A e. B -> (-. E.x e. B ph -> -. [A / x]ph))
6	5	a3d 75	. 2 |- (A e. B -> ([A / x]ph -> E.x e. B ph))
7	6	imp 350	1 |- ((A e. B /\ [A / x]ph) -> E.x e. B ph)

Syntax hints:  -. wn 2   -> wi 3   /\ wa 223   e. wcel 956  [wsbc 1168  A.wral 1642  E.wrex 1643

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-ral 1646  df-rex 1647  df-v 1808  df-sbc 1938

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