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

Theorem bijadc 810
Description: Combine antecedents into a single biconditional. This inference is reminiscent of jadc 794. (Contributed by Jim Kingdon, 4-May-2018.)
Hypotheses
Ref Expression
bijadc.1  |-  ( ph  ->  ( ps  ->  ch ) )
bijadc.2  |-  ( -. 
ph  ->  ( -.  ps  ->  ch ) )
Assertion
Ref Expression
bijadc  |-  (DECID  ps  ->  ( ( ph  <->  ps )  ->  ch ) )

Proof of Theorem bijadc
StepHypRef Expression
1 bi2 128 . . 3  |-  ( (
ph 
<->  ps )  ->  ( ps  ->  ph ) )
2 bijadc.1 . . 3  |-  ( ph  ->  ( ps  ->  ch ) )
31, 2syli 37 . 2  |-  ( (
ph 
<->  ps )  ->  ( ps  ->  ch ) )
4 bi1 116 . . . 4  |-  ( (
ph 
<->  ps )  ->  ( ph  ->  ps ) )
54con3d 594 . . 3  |-  ( (
ph 
<->  ps )  ->  ( -.  ps  ->  -.  ph )
)
6 bijadc.2 . . 3  |-  ( -. 
ph  ->  ( -.  ps  ->  ch ) )
75, 6syli 37 . 2  |-  ( (
ph 
<->  ps )  ->  ( -.  ps  ->  ch )
)
83, 7pm2.61ddc 792 1  |-  (DECID  ps  ->  ( ( ph  <->  ps )  ->  ch ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 103  DECID wdc 776
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663
This theorem depends on definitions:  df-bi 115  df-dc 777
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator