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Theorem ifpnot23 35869
Description: Negation of conditional logical operator. (Contributed by RP, 18-Apr-2020.)
Assertion
Ref Expression
ifpnot23  |-  ( -. if- ( ph ,  ps ,  ch )  <-> if- ( ph ,  -.  ps ,  -.  ch ) )

Proof of Theorem ifpnot23
StepHypRef Expression
1 ianor 490 . . . 4  |-  ( -.  ( ph  /\  ps ) 
<->  ( -.  ph  \/  -.  ps ) )
2 pm4.55 496 . . . 4  |-  ( -.  ( -.  ph  /\  ch )  <->  ( ph  \/  -.  ch ) )
31, 2anbi12i 701 . . 3  |-  ( ( -.  ( ph  /\  ps )  /\  -.  ( -.  ph  /\  ch )
)  <->  ( ( -. 
ph  \/  -.  ps )  /\  ( ph  \/  -.  ch ) ) )
4 ioran 492 . . 3  |-  ( -.  ( ( ph  /\  ps )  \/  ( -.  ph  /\  ch )
)  <->  ( -.  ( ph  /\  ps )  /\  -.  ( -.  ph  /\  ch ) ) )
5 dfifp4 1424 . . 3  |-  (if- (
ph ,  -.  ps ,  -.  ch )  <->  ( ( -.  ph  \/  -.  ps )  /\  ( ph  \/  -.  ch ) ) )
63, 4, 53bitr4i 280 . 2  |-  ( -.  ( ( ph  /\  ps )  \/  ( -.  ph  /\  ch )
)  <-> if- ( ph ,  -.  ps ,  -.  ch )
)
7 df-ifp 1421 . 2  |-  (if- (
ph ,  ps ,  ch )  <->  ( ( ph  /\ 
ps )  \/  ( -.  ph  /\  ch )
) )
86, 7xchnxbir 310 1  |-  ( -. if- ( ph ,  ps ,  ch )  <-> if- ( ph ,  -.  ps ,  -.  ch ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> 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:  ifpnotnotb  35870  ifpnorcor  35871  ifpnancor  35872  ifpnot23b  35873  ifpnot23c  35875  ifpnot23d  35876  ifpdfnan  35877  ifpdfxor  35878  ifpor123g  35899
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