Theorem ra4esbca 2538

Description: Existence form of ra4sbca 2537.

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 2536	. . . . 5 |- (A e. B -> (A.x e. B -. ph -> [A / x] -. ph))
2		sbcng 2495	. . . . 5 |- (A e. B -> ([A / x] -. ph <-> -. [A / x]ph))
3	1, 2	sylibd 219	. . . 4 |- (A e. B -> (A.x e. B -. ph -> -. [A / x]ph))
4		ralnex 2113	. . . 4 |- (A.x e. B -. ph <-> -. E.x e. B ph)
5	3, 4	syl5ibr 224	. . 3 |- (A e. B -> (-. E.x e. B ph -> -. [A / x]ph))
6	5	con4d 91	. 2 |- (A e. B -> ([A / x]ph -> E.x e. B ph))
7	6	imp 377	1 |- ((A e. B /\ [A / x]ph) -> E.x e. B ph)

Syntax hints:  -. wn 2   -> wi 3   /\ wa 240   e. wcel 1300  [wsbc 1534  A.wral 2105  E.wrex 2106

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-ral 2109  df-rex 2110  df-v 2294  df-sbc 2454

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