Theorem ra4esbca 1989

Description: Existence form of ra4sbca 1988.

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 1987	. . . . 5 |- (A e. B -> (A.x e. B -. ph -> [A / x] -. ph))
2		sbcng 1959	. . . . 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 1645	. . . 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 955  [wsbc 1166  A.wral 1637  E.wrex 1638

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-ral 1641  df-rex 1642  df-v 1803  df-sbc 1932

Copyright terms: Public domain