Theorem rspsbc2 38744

Description: rspsbc 3518 with two quantifying variables. This proof is rspsbc2VD 39090 automatically translated and minimized. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
rspsbc2	|- ( 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

Allowed substitution hints:   ph( x, y)   A( x)   B( y)   C( x, y)

Proof of Theorem rspsbc2

Step	Hyp	Ref	Expression
1		idd 24	. 2 |- ( A e. B -> ( C e. D -> C e. D ) )
2		rspsbc 3518	. . . 4 |- ( A e. B -> ( A. x e. B A. y e. D ph -> [. A / x ]. A. y e. D ph ) )
3	2	a1d 25	. . 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 3513	. . . 4 |- ( A e. B -> ( [. A / x ]. A. y e. D ph <-> A. y e. D [. A / x ]. ph ) )
5	4	biimpd 219	. . 3 |- ( A e. B -> ( [. A / x ]. A. y e. D ph -> A. y e. D [. A / x ]. ph ) )
6	3, 5	syl6d 75	. 2 |- ( A e. B -> ( C e. D -> ( A. x e. B A. y e. D ph -> A. y e. D [. A / x ]. ph ) ) )
7		rspsbc 3518	. 2 |- ( C e. D -> ( A. y e. D [. A / x ]. ph -> [. C / y ]. [. A / x ]. ph ) )
8	1, 6, 7	syl10 79	1 |- ( A e. B -> ( C e. D -> ( A. x e. B A. y e. D ph -> [. C / y ]. [. A / x ]. ph ) ) )

Syntax hints:   -> wi 4   e. wcel 1990   A.wral 2912   [.wsbc 3435

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-v 3202  df-sbc 3436

This theorem is referenced by:  tratrb  38746  tratrbVD  39097