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Theorem rspsbc2 36305
Description: rspsbc 3355 with two quantifying variables. This proof is rspsbc2VD 36665 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 24 . 2  |-  ( A  e.  B  ->  ( C  e.  D  ->  C  e.  D ) )
2 rspsbc 3355 . . . 4  |-  ( A  e.  B  ->  ( A. x  e.  B  A. y  e.  D  ph 
->  [. A  /  x ]. A. y  e.  D  ph ) )
32a1d 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 3351 . . . 4  |-  ( A  e.  B  ->  ( [. A  /  x ]. A. y  e.  D  ph  <->  A. y  e.  D  [. A  /  x ]. ph )
)
54biimpd 207 . . 3  |-  ( A  e.  B  ->  ( [. A  /  x ]. A. y  e.  D  ph 
->  A. y  e.  D  [. A  /  x ]. ph ) )
63, 5syl6d 68 . 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 3355 . 2  |-  ( C  e.  D  ->  ( A. y  e.  D  [. A  /  x ]. ph 
->  [. C  /  y ]. [. A  /  x ]. ph ) )
81, 6, 7syl10 72 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 setvar class
Syntax hints:    -> wi 4    e. wcel 1842   A.wral 2753   [.wsbc 3276
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ral 2758  df-v 3060  df-sbc 3277
This theorem is referenced by:  tratrb  36307  tratrbVD  36672
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