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Theorem anifp 1428
Description: The conditional operator is implied by the conjunction of its possible outputs. Dual statement of ifpor 1429. (Contributed by BJ, 30-Sep-2019.)
Assertion
Ref Expression
anifp  |-  ( ( ps  /\  ch )  -> if- ( ph ,  ps ,  ch ) )

Proof of Theorem anifp
StepHypRef Expression
1 olc 385 . . 3  |-  ( ps 
->  ( -.  ph  \/  ps ) )
2 olc 385 . . 3  |-  ( ch 
->  ( ph  \/  ch ) )
31, 2anim12i 568 . 2  |-  ( ( ps  /\  ch )  ->  ( ( -.  ph  \/  ps )  /\  ( ph  \/  ch ) ) )
4 dfifp4 1424 . 2  |-  (if- (
ph ,  ps ,  ch )  <->  ( ( -. 
ph  \/  ps )  /\  ( ph  \/  ch ) ) )
53, 4sylibr 215 1  |-  ( ( ps  /\  ch )  -> if- ( ph ,  ps ,  ch ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ 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:  bj-consensus  30941  bj-consensusALT  30942  axfrege58a  36107
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