Theorem ra4sbc2VD 16679

Description: Virtual deduction proof of ra4sbc2 5829. The following user's proof is completed by invoking mmj2's unify command and using mmj2's StepSelector to pick all remaining steps of the Metamath proof.

1::	|- . A e. B   ⊢   A e. B .
2::	|- . A e. B, C e. D   ⊢   C e. D .
3::	|- . A e. B, C e. D, A.x e. B A.y e. Dph   ⊢   A.x e. BA.y e. Dph .
4:1,3,?: e13 16616	|- . A e. B, C e. D, A.x e. B A.y e. Dph   ⊢   [A / x]A.y e. Dph .
5:1,4,?: e13 16616	|- . A e. B, C e. D, A.x e. B A.y e. Dph   ⊢   A.y e. D[A / x]ph .
6:2,5,?: e23 16623	|- . A e. B, C e. D, A.x e. B A.y e. Dph   ⊢   [C / y][A / x]ph .
7:6:	|- . A e. B, C e. D   ⊢   (A.x e. B A.y e. Dph -> [C / y][A / x]ph) .
8:7:	|- . A e. B   ⊢   (C e. D -> (A.x e. BA.y e. Dph -> [C / y][A / x]ph)) .
qed:8:	|- (A e. B -> (C e. D -> (A.x e. BA.y e. Dph -> [C / y][A / x]ph)))

Assertion

Ref	Expression
ra4sbc2VD	|- (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 ra4sbc2VD

Step	Hyp	Ref	Expression
1		idn2 16509	. . . . 5 |- . A e. B, C e. D   ⊢   C e. D .
2		idn1 16484	. . . . . 6 |- . A e. B   ⊢   A e. B .
3		idn3 16510	. . . . . . 7 |- . A e. B, C e. D, A.x e. B A.y e. D ph   ⊢   A.x e. B A.y e. D ph .
4		ra4sbc 2536	. . . . . . 7 |- (A e. B -> (A.x e. B A.y e. D ph -> [A / x]A.y e. D ph))
5	2, 3, 4	e13 16616	. . . . . 6 |- . A e. B, C e. D, A.x e. B A.y e. D ph   ⊢   [A / x]A.y e. D ph .
6		sbcralg 2531	. . . . . . 7 |- (A e. B -> ([A / x]A.y e. D ph <-> A.y e. D [A / x]ph))
7	6	biimpd 170	. . . . . 6 |- (A e. B -> ([A / x]A.y e. D ph -> A.y e. D [A / x]ph))
8	2, 5, 7	e13 16616	. . . . 5 |- . A e. B, C e. D, A.x e. B A.y e. D ph   ⊢   A.y e. D [A / x]ph .
9		ra4sbc 2536	. . . . 5 |- (C e. D -> (A.y e. D [A / x]ph -> [C / y][A / x]ph))
10	1, 8, 9	e23 16623	. . . 4 |- . A e. B, C e. D, A.x e. B A.y e. D ph   ⊢   [C / y][A / x]ph .
11	10	in3 16508	. . 3 |- . A e. B, C e. D   ⊢   (A.x e. B A.y e. D ph -> [C / y][A / x]ph) .
12	11	in2 16506	. 2 |- . A e. B   ⊢   (C e. D -> (A.x e. B A.y e. D ph -> [C / y][A / x]ph)) .
13	12	in1 16481	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 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-3an 860  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-ral 2109  df-v 2294  df-sbc 2454  df-vd1 16480  df-vd2 16489  df-vd3 16494

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