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Theorem rabxfr 4556
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 1312 . 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 452 . . 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 4555 . 2  |-  ( (  T.  /\  B  e.  D )  ->  ( C  e.  { x  e.  D  |  ph }  <->  B  e.  { y  e.  D  |  ps }
) )
91, 8mpan 651 1  |-  ( B  e.  D  ->  ( C  e.  { x  e.  D  |  ph }  <->  B  e.  { y  e.  D  |  ps }
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    T. wtru 1307    = wceq 1623    e. wcel 1684   F/_wnfc 2406   {crab 2547
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rab 2552  df-v 2790
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