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Theorem rspsbc2 31237
Description: rspsbc 3274 with two quantifying variables. This proof is rspsbc2VD 31588 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 3274 . . . 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 3268 . . . 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 69 . 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 3274 . 2  |-  ( C  e.  D  ->  ( A. y  e.  D  [. A  /  x ]. ph 
->  [. C  /  y ]. [. A  /  x ]. ph ) )
81, 6, 7syl10 73 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 1756   A.wral 2713   [.wsbc 3184
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ral 2718  df-v 2972  df-sbc 3185
This theorem is referenced by:  tratrb  31239  tratrbVD  31594
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