Theorem ra4sbc2 5829

Description: ra4sbc 2536 with two quantifying variables. This proof is ra4sbc2VD 16679 automatically translated and minimized. (Contributed by Alan Sare, 31-Dec-2011.)

Assertion

Ref	Expression
ra4sbc2	|- (A e. B -> (C e. D -> (A.x e. B A.y e. D ph -> [C / y][A / x]ph)))

Distinct variable groups:   y,A   x,B   x,D,y

Proof of Theorem ra4sbc2

Step	Hyp	Ref	Expression
1		idd 75	. 2 |- (A e. B -> (C e. D -> C e. D))
2		ra4sbc 2536	. . . 4 |- (A e. B -> (A.x e. B A.y e. D ph -> [A / x]A.y e. D ph))
3	2	a1d 15	. . 3 |- (A e. B -> (C e. D -> (A.x e. B A.y e. D ph -> [A / x]A.y e. D ph)))
4		sbcralg 2531	. . . 4 |- (A e. B -> ([A / x]A.y e. D ph <-> A.y e. D [A / x]ph))
5	4	biimpd 170	. . 3 |- (A e. B -> ([A / x]A.y e. D ph -> A.y e. D [A / x]ph))
6	3, 5	syl6d 67	. 2 |- (A e. B -> (C e. D -> (A.x e. B A.y e. D ph -> A.y e. D [A / x]ph)))
7		ra4sbc 2536	. 2 |- (C e. D -> (A.y e. D [A / x]ph -> [C / y][A / x]ph))
8	1, 6, 7	ee23 1276	1 |- (A e. B -> (C e. D -> (A.x e. B A.y e. D ph -> [C / y][A / x]ph)))

Syntax hints:   -> wi 3   e. wcel 1300  [wsbc 1534  A.wral 2105

This theorem is referenced by:  tratrb 5831  tratrbVD 16685

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-v 2294  df-sbc 2454

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