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Theorem ifpbi12 35579
Description: Equivalence theorem for conditional logical operators. (Contributed by RP, 15-Apr-2020.)
Assertion
Ref Expression
ifpbi12  |-  ( ( ( ph  <->  ps )  /\  ( ch  <->  th )
)  ->  (if- ( ph ,  ch ,  ta ) 
<-> if- ( ps ,  th ,  ta ) ) )

Proof of Theorem ifpbi12
StepHypRef Expression
1 imbi12 320 . . . 4  |-  ( (
ph 
<->  ps )  ->  (
( ch  <->  th )  ->  ( ( ph  ->  ch )  <->  ( ps  ->  th ) ) ) )
21imp 427 . . 3  |-  ( ( ( ph  <->  ps )  /\  ( ch  <->  th )
)  ->  ( ( ph  ->  ch )  <->  ( ps  ->  th ) ) )
3 simpl 455 . . . . 5  |-  ( ( ( ph  <->  ps )  /\  ( ch  <->  th )
)  ->  ( ph  <->  ps ) )
43notbid 292 . . . 4  |-  ( ( ( ph  <->  ps )  /\  ( ch  <->  th )
)  ->  ( -.  ph  <->  -. 
ps ) )
54imbi1d 315 . . 3  |-  ( ( ( ph  <->  ps )  /\  ( ch  <->  th )
)  ->  ( ( -.  ph  ->  ta )  <->  ( -.  ps  ->  ta ) ) )
62, 5anbi12d 709 . 2  |-  ( ( ( ph  <->  ps )  /\  ( ch  <->  th )
)  ->  ( (
( ph  ->  ch )  /\  ( -.  ph  ->  ta ) )  <->  ( ( ps  ->  th )  /\  ( -.  ps  ->  ta )
) ) )
7 dfifp2 1388 . 2  |-  (if- (
ph ,  ch ,  ta )  <->  ( ( ph  ->  ch )  /\  ( -.  ph  ->  ta )
) )
8 dfifp2 1388 . 2  |-  (if- ( ps ,  th ,  ta )  <->  ( ( ps 
->  th )  /\  ( -.  ps  ->  ta )
) )
96, 7, 83bitr4g 288 1  |-  ( ( ( ph  <->  ps )  /\  ( ch  <->  th )
)  ->  (if- ( ph ,  ch ,  ta ) 
<-> if- ( ps ,  th ,  ta ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367  if-wif 1386
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-ifp 1387
This theorem is referenced by: (None)
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