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Theorem ifp1bi 36140
Description: Substitute the first element of conditional logical operator. (Contributed by RP, 20-Apr-2020.)
Assertion
Ref Expression
ifp1bi  |-  ( (if- ( ph ,  ch ,  th )  <-> if- ( ps ,  ch ,  th )
)  <->  ( ( ( ( ph  ->  ps )  \/  ( ch  ->  th ) )  /\  ( ( ph  ->  ps )  \/  ( th 
->  ch ) ) )  /\  ( ( ( ps  ->  ph )  \/  ( ch  ->  th )
)  /\  ( ( ps  ->  ph )  \/  ( th  ->  ch ) ) ) ) )

Proof of Theorem ifp1bi
StepHypRef Expression
1 dfbi2 633 . 2  |-  ( (if- ( ph ,  ch ,  th )  <-> if- ( ps ,  ch ,  th )
)  <->  ( (if- (
ph ,  ch ,  th )  -> if- ( ps ,  ch ,  th ) )  /\  (if- ( ps ,  ch ,  th )  -> if- ( ph ,  ch ,  th )
) ) )
2 ifpim1g 36139 . . . 4  |-  ( (if- ( ph ,  ch ,  th )  -> if- ( ps ,  ch ,  th ) )  <->  ( (
( ps  ->  ph )  \/  ( th  ->  ch ) )  /\  (
( ph  ->  ps )  \/  ( ch  ->  th )
) ) )
3 ancom 452 . . . 4  |-  ( ( ( ( ps  ->  ph )  \/  ( th 
->  ch ) )  /\  ( ( ph  ->  ps )  \/  ( ch 
->  th ) ) )  <-> 
( ( ( ph  ->  ps )  \/  ( ch  ->  th ) )  /\  ( ( ps  ->  ph )  \/  ( th 
->  ch ) ) ) )
42, 3bitri 253 . . 3  |-  ( (if- ( ph ,  ch ,  th )  -> if- ( ps ,  ch ,  th ) )  <->  ( (
( ph  ->  ps )  \/  ( ch  ->  th )
)  /\  ( ( ps  ->  ph )  \/  ( th  ->  ch ) ) ) )
5 ifpim1g 36139 . . 3  |-  ( (if- ( ps ,  ch ,  th )  -> if- ( ph ,  ch ,  th )
)  <->  ( ( (
ph  ->  ps )  \/  ( th  ->  ch ) )  /\  (
( ps  ->  ph )  \/  ( ch  ->  th )
) ) )
64, 5anbi12i 702 . 2  |-  ( ( (if- ( ph ,  ch ,  th )  -> if- ( ps ,  ch ,  th ) )  /\  (if- ( ps ,  ch ,  th )  -> if- ( ph ,  ch ,  th )
) )  <->  ( (
( ( ph  ->  ps )  \/  ( ch 
->  th ) )  /\  ( ( ps  ->  ph )  \/  ( th 
->  ch ) ) )  /\  ( ( (
ph  ->  ps )  \/  ( th  ->  ch ) )  /\  (
( ps  ->  ph )  \/  ( ch  ->  th )
) ) ) )
7 an42 833 . 2  |-  ( ( ( ( ( ph  ->  ps )  \/  ( ch  ->  th ) )  /\  ( ( ps  ->  ph )  \/  ( th 
->  ch ) ) )  /\  ( ( (
ph  ->  ps )  \/  ( th  ->  ch ) )  /\  (
( ps  ->  ph )  \/  ( ch  ->  th )
) ) )  <->  ( (
( ( ph  ->  ps )  \/  ( ch 
->  th ) )  /\  ( ( ph  ->  ps )  \/  ( th 
->  ch ) ) )  /\  ( ( ( ps  ->  ph )  \/  ( ch  ->  th )
)  /\  ( ( ps  ->  ph )  \/  ( th  ->  ch ) ) ) ) )
81, 6, 73bitri 275 1  |-  ( (if- ( ph ,  ch ,  th )  <-> if- ( ps ,  ch ,  th )
)  <->  ( ( ( ( ph  ->  ps )  \/  ( ch  ->  th ) )  /\  ( ( ph  ->  ps )  \/  ( th 
->  ch ) ) )  /\  ( ( ( ps  ->  ph )  \/  ( ch  ->  th )
)  /\  ( ( ps  ->  ph )  \/  ( th  ->  ch ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    \/ wo 370    /\ wa 371  if-wif 1424
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 189  df-or 372  df-an 373  df-ifp 1425
This theorem is referenced by: (None)
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