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Theorem ifpbi1b 36117
Description: When the first variable is irrelevant, it can be replaced. (Contributed by RP, 25-Apr-2020.)
Assertion
Ref Expression
ifpbi1b  |-  (if- (
ph ,  ch ,  ch )  <-> if- ( ps ,  ch ,  ch ) )

Proof of Theorem ifpbi1b
StepHypRef Expression
1 id 22 . . . . 5  |-  ( ch 
->  ch )
21olci 392 . . . 4  |-  ( (
ph  ->  -.  ps )  \/  ( ch  ->  ch ) )
31olci 392 . . . 4  |-  ( ( ps  ->  ph )  \/  ( ch  ->  ch ) )
42, 3pm3.2i 456 . . 3  |-  ( ( ( ph  ->  -.  ps )  \/  ( ch  ->  ch ) )  /\  ( ( ps 
->  ph )  \/  ( ch  ->  ch ) ) )
51olci 392 . . . 4  |-  ( (
ph  ->  ps )  \/  ( ch  ->  ch ) )
61olci 392 . . . 4  |-  ( ( -.  ps  ->  ph )  \/  ( ch  ->  ch ) )
75, 6pm3.2i 456 . . 3  |-  ( ( ( ph  ->  ps )  \/  ( ch  ->  ch ) )  /\  ( ( -.  ps  ->  ph )  \/  ( ch  ->  ch ) ) )
8 ifpim123g 36114 . . 3  |-  ( (if- ( ph ,  ch ,  ch )  -> if- ( ps ,  ch ,  ch ) )  <->  ( (
( ( ph  ->  -. 
ps )  \/  ( ch  ->  ch ) )  /\  ( ( ps 
->  ph )  \/  ( ch  ->  ch ) ) )  /\  ( ( ( ph  ->  ps )  \/  ( ch  ->  ch ) )  /\  ( ( -.  ps  ->  ph )  \/  ( ch  ->  ch ) ) ) ) )
94, 7, 8mpbir2an 928 . 2  |-  (if- (
ph ,  ch ,  ch )  -> if- ( ps ,  ch ,  ch ) )
101olci 392 . . . 4  |-  ( ( ps  ->  -.  ph )  \/  ( ch  ->  ch ) )
1110, 5pm3.2i 456 . . 3  |-  ( ( ( ps  ->  -.  ph )  \/  ( ch 
->  ch ) )  /\  ( ( ph  ->  ps )  \/  ( ch 
->  ch ) ) )
121olci 392 . . . 4  |-  ( ( -.  ph  ->  ps )  \/  ( ch  ->  ch ) )
133, 12pm3.2i 456 . . 3  |-  ( ( ( ps  ->  ph )  \/  ( ch  ->  ch ) )  /\  (
( -.  ph  ->  ps )  \/  ( ch 
->  ch ) ) )
14 ifpim123g 36114 . . 3  |-  ( (if- ( ps ,  ch ,  ch )  -> if- ( ph ,  ch ,  ch )
)  <->  ( ( ( ( ps  ->  -.  ph )  \/  ( ch 
->  ch ) )  /\  ( ( ph  ->  ps )  \/  ( ch 
->  ch ) ) )  /\  ( ( ( ps  ->  ph )  \/  ( ch  ->  ch ) )  /\  (
( -.  ph  ->  ps )  \/  ( ch 
->  ch ) ) ) ) )
1511, 13, 14mpbir2an 928 . 2  |-  (if- ( ps ,  ch ,  ch )  -> if- ( ph ,  ch ,  ch )
)
169, 15impbii 190 1  |-  (if- (
ph ,  ch ,  ch )  <-> if- ( ps ,  ch ,  ch ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    \/ wo 369    /\ wa 370  if-wif 1420
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 188  df-or 371  df-an 372  df-ifp 1421
This theorem is referenced by: (None)
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