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Theorem rspsbc2 27961
Description: rspsbc 3182 with two quantifying variables. This proof is rspsbc2VD 28308 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
StepHypRef Expression
1 idd 22 . 2  |-  ( A  e.  B  ->  ( C  e.  D  ->  C  e.  D ) )
2 rspsbc 3182 . . . 4  |-  ( A  e.  B  ->  ( A. x  e.  B  A. y  e.  D  ph 
->  [. A  /  x ]. A. y  e.  D  ph ) )
32a1d 23 . . 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 3178 . . . 4  |-  ( A  e.  B  ->  ( [. A  /  x ]. A. y  e.  D  ph  <->  A. y  e.  D  [. A  /  x ]. ph )
)
54biimpd 199 . . 3  |-  ( A  e.  B  ->  ( [. A  /  x ]. A. y  e.  D  ph 
->  A. y  e.  D  [. A  /  x ]. ph ) )
63, 5syl6d 66 . 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 3182 . 2  |-  ( C  e.  D  ->  ( A. y  e.  D  [. A  /  x ]. ph 
->  [. C  /  y ]. [. A  /  x ]. ph ) )
81, 6, 7ee23 1370 1  |-  ( A  e.  B  ->  ( C  e.  D  ->  ( A. x  e.  B  A. y  e.  D  ph 
->  [. C  /  y ]. [. A  /  x ]. ph ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1717   A.wral 2649   [.wsbc 3104
This theorem is referenced by:  tratrb  27963  tratrbVD  28314
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ral 2654  df-v 2901  df-sbc 3105
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