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Theorem ifpbi2 36036
Description: Equivalence theorem for conditional logical operators. (Contributed by RP, 14-Apr-2020.)
Assertion
Ref Expression
ifpbi2  |-  ( (
ph 
<->  ps )  ->  (if- ( ch ,  ph ,  th )  <-> if- ( ch ,  ps ,  th ) ) )

Proof of Theorem ifpbi2
StepHypRef Expression
1 imbi2 326 . . 3  |-  ( (
ph 
<->  ps )  ->  (
( ch  ->  ph )  <->  ( ch  ->  ps )
) )
21anbi1d 710 . 2  |-  ( (
ph 
<->  ps )  ->  (
( ( ch  ->  ph )  /\  ( -. 
ch  ->  th ) )  <->  ( ( ch  ->  ps )  /\  ( -.  ch  ->  th ) ) ) )
3 dfifp2 1423 . 2  |-  (if- ( ch ,  ph ,  th )  <->  ( ( ch 
->  ph )  /\  ( -.  ch  ->  th )
) )
4 dfifp2 1423 . 2  |-  (if- ( ch ,  ps ,  th )  <->  ( ( ch 
->  ps )  /\  ( -.  ch  ->  th )
) )
52, 3, 43bitr4g 292 1  |-  ( (
ph 
<->  ps )  ->  (if- ( ch ,  ph ,  th )  <-> if- ( ch ,  ps ,  th ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 188    /\ wa 371  if-wif 1421
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 1422
This theorem is referenced by:  ifpnot23b  36052
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