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Theorem ifpim1g 38131
Description: Implication of conditional logical operators. (Contributed by RP, 18-Apr-2020.)
Assertion
Ref Expression
ifpim1g  |-  ( (if- ( ph ,  ch ,  th )  -> if- ( ps ,  ch ,  th ) )  <->  ( (
( ps  ->  ph )  \/  ( th  ->  ch ) )  /\  (
( ph  ->  ps )  \/  ( ch  ->  th )
) ) )

Proof of Theorem ifpim1g
StepHypRef Expression
1 ifpim123g 38119 . 2  |-  ( (if- ( ph ,  ch ,  th )  -> if- ( ps ,  ch ,  th ) )  <->  ( (
( ( ph  ->  -. 
ps )  \/  ( ch  ->  ch ) )  /\  ( ( ps 
->  ph )  \/  ( th  ->  ch ) ) )  /\  ( ( ( ph  ->  ps )  \/  ( ch  ->  th ) )  /\  ( ( -.  ps  ->  ph )  \/  ( th  ->  th ) ) ) ) )
2 id 22 . . . . . 6  |-  ( ch 
->  ch )
32olci 389 . . . . 5  |-  ( (
ph  ->  -.  ps )  \/  ( ch  ->  ch ) )
43biantrur 504 . . . 4  |-  ( ( ( ps  ->  ph )  \/  ( th  ->  ch ) )  <->  ( (
( ph  ->  -.  ps )  \/  ( ch  ->  ch ) )  /\  ( ( ps  ->  ph )  \/  ( th 
->  ch ) ) ) )
54bicomi 202 . . 3  |-  ( ( ( ( ph  ->  -. 
ps )  \/  ( ch  ->  ch ) )  /\  ( ( ps 
->  ph )  \/  ( th  ->  ch ) ) )  <->  ( ( ps 
->  ph )  \/  ( th  ->  ch ) ) )
6 id 22 . . . . . 6  |-  ( th 
->  th )
76olci 389 . . . . 5  |-  ( ( -.  ps  ->  ph )  \/  ( th  ->  th )
)
87biantru 503 . . . 4  |-  ( ( ( ph  ->  ps )  \/  ( ch  ->  th ) )  <->  ( (
( ph  ->  ps )  \/  ( ch  ->  th )
)  /\  ( ( -.  ps  ->  ph )  \/  ( th  ->  th )
) ) )
98bicomi 202 . . 3  |-  ( ( ( ( ph  ->  ps )  \/  ( ch 
->  th ) )  /\  ( ( -.  ps  ->  ph )  \/  ( th  ->  th ) ) )  <-> 
( ( ph  ->  ps )  \/  ( ch 
->  th ) ) )
105, 9anbi12i 695 . 2  |-  ( ( ( ( ( ph  ->  -.  ps )  \/  ( ch  ->  ch ) )  /\  (
( ps  ->  ph )  \/  ( th  ->  ch ) ) )  /\  ( ( ( ph  ->  ps )  \/  ( ch  ->  th ) )  /\  ( ( -.  ps  ->  ph )  \/  ( th  ->  th ) ) ) )  <->  ( ( ( ps  ->  ph )  \/  ( th  ->  ch ) )  /\  (
( ph  ->  ps )  \/  ( ch  ->  th )
) ) )
111, 10bitri 249 1  |-  ( (if- ( ph ,  ch ,  th )  -> if- ( ps ,  ch ,  th ) )  <->  ( (
( ps  ->  ph )  \/  ( th  ->  ch ) )  /\  (
( ph  ->  ps )  \/  ( ch  ->  th )
) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 366    /\ wa 367  if-wif 1382
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 1383
This theorem is referenced by:  ifp1bi  38142
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