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Theorem rabxfr 4614
Description: Class builder membership after substituting an expression  A (containing  y) for  x in the class expression  ph. (Contributed by NM, 10-Jun-2005.)
Hypotheses
Ref Expression
rabxfr.1  |-  F/_ y B
rabxfr.2  |-  F/_ y C
rabxfr.3  |-  ( y  e.  D  ->  A  e.  D )
rabxfr.4  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
rabxfr.5  |-  ( y  =  B  ->  A  =  C )
Assertion
Ref Expression
rabxfr  |-  ( B  e.  D  ->  ( C  e.  { x  e.  D  |  ph }  <->  B  e.  { y  e.  D  |  ps }
) )
Distinct variable groups:    x, A    x, y, D    ph, y    ps, x
Allowed substitution hints:    ph( x)    ps( y)    A( y)    B( x, y)    C( x, y)

Proof of Theorem rabxfr
StepHypRef Expression
1 tru 1374 . 2  |- T.
2 rabxfr.1 . . 3  |-  F/_ y B
3 rabxfr.2 . . 3  |-  F/_ y C
4 rabxfr.3 . . . 4  |-  ( y  e.  D  ->  A  e.  D )
54adantl 466 . . 3  |-  ( ( T.  /\  y  e.  D )  ->  A  e.  D )
6 rabxfr.4 . . 3  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
7 rabxfr.5 . . 3  |-  ( y  =  B  ->  A  =  C )
82, 3, 5, 6, 7rabxfrd 4613 . 2  |-  ( ( T.  /\  B  e.  D )  ->  ( C  e.  { x  e.  D  |  ph }  <->  B  e.  { y  e.  D  |  ps }
) )
91, 8mpan 670 1  |-  ( B  e.  D  ->  ( C  e.  { x  e.  D  |  ph }  <->  B  e.  { y  e.  D  |  ps }
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1370   T. wtru 1371    e. wcel 1758   F/_wnfc 2599   {crab 2799
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ral 2800  df-rab 2804  df-v 3072
This theorem is referenced by: (None)
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