Users' Mathboxes Mathbox for Alan Sare < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  rspsbc2 Unicode version

Theorem rspsbc2 28596
Description: rspsbc 3082 with two quantifying variables. This proof is rspsbc2VD 28947 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 21 . 2  |-  ( A  e.  B  ->  ( C  e.  D  ->  C  e.  D ) )
2 rspsbc 3082 . . . 4  |-  ( A  e.  B  ->  ( A. x  e.  B  A. y  e.  D  ph 
->  [. A  /  x ]. A. y  e.  D  ph ) )
32a1d 22 . . 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 3078 . . . 4  |-  ( A  e.  B  ->  ( [. A  /  x ]. A. y  e.  D  ph  <->  A. y  e.  D  [. A  /  x ]. ph )
)
54biimpd 198 . . 3  |-  ( A  e.  B  ->  ( [. A  /  x ]. A. y  e.  D  ph 
->  A. y  e.  D  [. A  /  x ]. ph ) )
63, 5syl6d 64 . 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 3082 . 2  |-  ( C  e.  D  ->  ( A. y  e.  D  [. A  /  x ]. ph 
->  [. C  /  y ]. [. A  /  x ]. ph ) )
81, 6, 7ee23 1354 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 1696   A.wral 2556   [.wsbc 3004
This theorem is referenced by:  tratrb  28598  tratrbVD  28953
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-v 2803  df-sbc 3005
  Copyright terms: Public domain W3C validator