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Theorem 2gencl 3109
Description: Implicit substitution for class with embedded variable. (Contributed by NM, 17-May-1996.)
Hypotheses
Ref Expression
2gencl.1  |-  ( C  e.  S  <->  E. x  e.  R  A  =  C )
2gencl.2  |-  ( D  e.  S  <->  E. y  e.  R  B  =  D )
2gencl.3  |-  ( A  =  C  ->  ( ph 
<->  ps ) )
2gencl.4  |-  ( B  =  D  ->  ( ps 
<->  ch ) )
2gencl.5  |-  ( ( x  e.  R  /\  y  e.  R )  ->  ph )
Assertion
Ref Expression
2gencl  |-  ( ( C  e.  S  /\  D  e.  S )  ->  ch )
Distinct variable groups:    x, y    x, R    ps, x    y, C    y, S    ch, y
Allowed substitution hints:    ph( x, y)    ps( y)    ch( x)    A( x, y)    B( x, y)    C( x)    D( x, y)    R( y)    S( x)

Proof of Theorem 2gencl
StepHypRef Expression
1 2gencl.2 . . . 4  |-  ( D  e.  S  <->  E. y  e.  R  B  =  D )
2 df-rex 2779 . . . 4  |-  ( E. y  e.  R  B  =  D  <->  E. y ( y  e.  R  /\  B  =  D ) )
31, 2bitri 252 . . 3  |-  ( D  e.  S  <->  E. y
( y  e.  R  /\  B  =  D
) )
4 2gencl.4 . . . 4  |-  ( B  =  D  ->  ( ps 
<->  ch ) )
54imbi2d 317 . . 3  |-  ( B  =  D  ->  (
( C  e.  S  ->  ps )  <->  ( C  e.  S  ->  ch )
) )
6 2gencl.1 . . . . . 6  |-  ( C  e.  S  <->  E. x  e.  R  A  =  C )
7 df-rex 2779 . . . . . 6  |-  ( E. x  e.  R  A  =  C  <->  E. x ( x  e.  R  /\  A  =  C ) )
86, 7bitri 252 . . . . 5  |-  ( C  e.  S  <->  E. x
( x  e.  R  /\  A  =  C
) )
9 2gencl.3 . . . . . 6  |-  ( A  =  C  ->  ( ph 
<->  ps ) )
109imbi2d 317 . . . . 5  |-  ( A  =  C  ->  (
( y  e.  R  ->  ph )  <->  ( y  e.  R  ->  ps )
) )
11 2gencl.5 . . . . . 6  |-  ( ( x  e.  R  /\  y  e.  R )  ->  ph )
1211ex 435 . . . . 5  |-  ( x  e.  R  ->  (
y  e.  R  ->  ph ) )
138, 10, 12gencl 3108 . . . 4  |-  ( C  e.  S  ->  (
y  e.  R  ->  ps ) )
1413com12 32 . . 3  |-  ( y  e.  R  ->  ( C  e.  S  ->  ps ) )
153, 5, 14gencl 3108 . 2  |-  ( D  e.  S  ->  ( C  e.  S  ->  ch ) )
1615impcom 431 1  |-  ( ( C  e.  S  /\  D  e.  S )  ->  ch )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437   E.wex 1659    e. wcel 1867   E.wrex 2774
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748
This theorem depends on definitions:  df-bi 188  df-an 372  df-ex 1660  df-rex 2779
This theorem is referenced by:  3gencl  3110  axaddrcl  9565  axmulrcl  9567  axpre-lttri  9578  axpre-mulgt0  9581  uzin2  13375
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