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Theorem 3gencl 2831
Description: Implicit substitution for class with embedded variable. (Contributed by NM, 17-May-1996.)
Hypotheses
Ref Expression
3gencl.1  |-  ( D  e.  S  <->  E. x  e.  R  A  =  D )
3gencl.2  |-  ( F  e.  S  <->  E. y  e.  R  B  =  F )
3gencl.3  |-  ( G  e.  S  <->  E. z  e.  R  C  =  G )
3gencl.4  |-  ( A  =  D  ->  ( ph 
<->  ps ) )
3gencl.5  |-  ( B  =  F  ->  ( ps 
<->  ch ) )
3gencl.6  |-  ( C  =  G  ->  ( ch 
<->  th ) )
3gencl.7  |-  ( ( x  e.  R  /\  y  e.  R  /\  z  e.  R )  ->  ph )
Assertion
Ref Expression
3gencl  |-  ( ( D  e.  S  /\  F  e.  S  /\  G  e.  S )  ->  th )
Distinct variable groups:    x, y,
z    y, D, z    z, F    x, R, y    y, S, z    ps, x    ch, y    th, z
Allowed substitution hints:    ph( x, y, z)    ps( y, z)    ch( x, z)    th( x, y)    A( x, y, z)    B( x, y, z)    C( x, y, z)    D( x)    R( z)    S( x)    F( x, y)    G( x, y, z)

Proof of Theorem 3gencl
StepHypRef Expression
1 3gencl.3 . . . . 5  |-  ( G  e.  S  <->  E. z  e.  R  C  =  G )
2 df-rex 2562 . . . . 5  |-  ( E. z  e.  R  C  =  G  <->  E. z ( z  e.  R  /\  C  =  G ) )
31, 2bitri 240 . . . 4  |-  ( G  e.  S  <->  E. z
( z  e.  R  /\  C  =  G
) )
4 3gencl.6 . . . . 5  |-  ( C  =  G  ->  ( ch 
<->  th ) )
54imbi2d 307 . . . 4  |-  ( C  =  G  ->  (
( ( D  e.  S  /\  F  e.  S )  ->  ch ) 
<->  ( ( D  e.  S  /\  F  e.  S )  ->  th )
) )
6 3gencl.1 . . . . . 6  |-  ( D  e.  S  <->  E. x  e.  R  A  =  D )
7 3gencl.2 . . . . . 6  |-  ( F  e.  S  <->  E. y  e.  R  B  =  F )
8 3gencl.4 . . . . . . 7  |-  ( A  =  D  ->  ( ph 
<->  ps ) )
98imbi2d 307 . . . . . 6  |-  ( A  =  D  ->  (
( z  e.  R  ->  ph )  <->  ( z  e.  R  ->  ps )
) )
10 3gencl.5 . . . . . . 7  |-  ( B  =  F  ->  ( ps 
<->  ch ) )
1110imbi2d 307 . . . . . 6  |-  ( B  =  F  ->  (
( z  e.  R  ->  ps )  <->  ( z  e.  R  ->  ch )
) )
12 3gencl.7 . . . . . . 7  |-  ( ( x  e.  R  /\  y  e.  R  /\  z  e.  R )  ->  ph )
13123expia 1153 . . . . . 6  |-  ( ( x  e.  R  /\  y  e.  R )  ->  ( z  e.  R  ->  ph ) )
146, 7, 9, 11, 132gencl 2830 . . . . 5  |-  ( ( D  e.  S  /\  F  e.  S )  ->  ( z  e.  R  ->  ch ) )
1514com12 27 . . . 4  |-  ( z  e.  R  ->  (
( D  e.  S  /\  F  e.  S
)  ->  ch )
)
163, 5, 15gencl 2829 . . 3  |-  ( G  e.  S  ->  (
( D  e.  S  /\  F  e.  S
)  ->  th )
)
1716com12 27 . 2  |-  ( ( D  e.  S  /\  F  e.  S )  ->  ( G  e.  S  ->  th ) )
18173impia 1148 1  |-  ( ( D  e.  S  /\  F  e.  S  /\  G  e.  S )  ->  th )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934   E.wex 1531    = wceq 1632    e. wcel 1696   E.wrex 2557
This theorem is referenced by:  axpre-lttrn  8804  axpre-ltadd  8805
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
This theorem depends on definitions:  df-bi 177  df-an 360  df-3an 936  df-ex 1532  df-rex 2562
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