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Theorem ifpbi1 36035
Description: Equivalence theorem for conditional logical operators. (Contributed by RP, 14-Apr-2020.)
Assertion
Ref Expression
ifpbi1  |-  ( (
ph 
<->  ps )  ->  (if- ( ph ,  ch ,  th )  <-> if- ( ps ,  ch ,  th ) ) )

Proof of Theorem ifpbi1
StepHypRef Expression
1 imbi1 324 . . 3  |-  ( (
ph 
<->  ps )  ->  (
( ph  ->  ch )  <->  ( ps  ->  ch )
) )
2 notbi 296 . . . . 5  |-  ( (
ph 
<->  ps )  <->  ( -.  ph  <->  -. 
ps ) )
32biimpi 197 . . . 4  |-  ( (
ph 
<->  ps )  ->  ( -.  ph  <->  -.  ps )
)
43imbi1d 318 . . 3  |-  ( (
ph 
<->  ps )  ->  (
( -.  ph  ->  th )  <->  ( -.  ps  ->  th ) ) )
51, 4anbi12d 715 . 2  |-  ( (
ph 
<->  ps )  ->  (
( ( ph  ->  ch )  /\  ( -. 
ph  ->  th ) )  <->  ( ( ps  ->  ch )  /\  ( -.  ps  ->  th ) ) ) )
6 dfifp2 1422 . 2  |-  (if- (
ph ,  ch ,  th )  <->  ( ( ph  ->  ch )  /\  ( -.  ph  ->  th )
) )
7 dfifp2 1422 . 2  |-  (if- ( ps ,  ch ,  th )  <->  ( ( ps 
->  ch )  /\  ( -.  ps  ->  th )
) )
85, 6, 73bitr4g 291 1  |-  ( (
ph 
<->  ps )  ->  (if- ( ph ,  ch ,  th )  <-> if- ( ps ,  ch ,  th ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    /\ 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:  ifpimim  36067  axfrege52a  36304
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