ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  gencl Unicode version

Theorem gencl 2651
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 152 . . . 4  |-  ( A  =  B  ->  ( ch  ->  ps ) )
54impcom 123 . . 3  |-  ( ( ch  /\  A  =  B )  ->  ps )
65exlimiv 1534 . 2  |-  ( E. x ( ch  /\  A  =  B )  ->  ps )
71, 6sylbi 119 1  |-  ( th 
->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1289   E.wex 1426
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-gen 1383  ax-ie2 1428  ax-17 1464
This theorem depends on definitions:  df-bi 115
This theorem is referenced by:  2gencl  2652  3gencl  2653  axprecex  7394  axpre-ltirr  7396
  Copyright terms: Public domain W3C validator