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Theorem ifpdfan2 35805
Description: Define and with conditional logic operator. (Contributed by RP, 25-Apr-2020.)
Assertion
Ref Expression
ifpdfan2  |-  ( (
ph  /\  ps )  <-> if- (
ph ,  ps ,  ph ) )

Proof of Theorem ifpdfan2
StepHypRef Expression
1 id 23 . . . 4  |-  ( ph  ->  ph )
21notnoti 126 . . 3  |-  -.  -.  ( ph  ->  ph )
32biorfi 408 . 2  |-  ( (
ph  /\  ps )  <->  ( ( ph  /\  ps )  \/  -.  ( ph  ->  ph ) ) )
4 dfifp6 1426 . 2  |-  (if- (
ph ,  ps ,  ph )  <->  ( ( ph  /\ 
ps )  \/  -.  ( ph  ->  ph ) ) )
53, 4bitr4i 255 1  |-  ( (
ph  /\  ps )  <-> if- (
ph ,  ps ,  ph ) )
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:  ifpancor  35806
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