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Theorem gencl 2692
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 153 . . . 4  |-  ( A  =  B  ->  ( ch  ->  ps ) )
54impcom 124 . . 3  |-  ( ( ch  /\  A  =  B )  ->  ps )
65exlimiv 1562 . 2  |-  ( E. x ( ch  /\  A  =  B )  ->  ps )
71, 6sylbi 120 1  |-  ( th 
->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1316   E.wex 1453
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-gen 1410  ax-ie2 1455  ax-17 1491
This theorem depends on definitions:  df-bi 116
This theorem is referenced by:  2gencl  2693  3gencl  2694  axprecex  7656  axpre-ltirr  7658
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