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Theorem gencl 2829
Description: Implicit substitution for class with embedded variable. (Contributed by NM, 17-May-1996.)
Hypotheses
Ref Expression
gencl.1  |-  ( th  <->  E. x ( ch  /\  A  =  B )
)
gencl.2  |-  ( A  =  B  ->  ( ph 
<->  ps ) )
gencl.3  |-  ( ch 
->  ph )
Assertion
Ref Expression
gencl  |-  ( th 
->  ps )
Distinct variable group:    ps, x
Allowed substitution hints:    ph( x)    ch( x)    th( x)    A( x)    B( x)

Proof of Theorem gencl
StepHypRef Expression
1 gencl.1 . 2  |-  ( th  <->  E. x ( ch  /\  A  =  B )
)
2 gencl.3 . . . . 5  |-  ( ch 
->  ph )
3 gencl.2 . . . . 5  |-  ( A  =  B  ->  ( ph 
<->  ps ) )
42, 3syl5ib 210 . . . 4  |-  ( A  =  B  ->  ( ch  ->  ps ) )
54impcom 419 . . 3  |-  ( ( ch  /\  A  =  B )  ->  ps )
65exlimiv 1624 . 2  |-  ( E. x ( ch  /\  A  =  B )  ->  ps )
71, 6sylbi 187 1  |-  ( th 
->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   E.wex 1531    = wceq 1632
This theorem is referenced by:  2gencl  2830  3gencl  2831  indpi  8547  axrrecex  8801
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-ex 1532
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