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Theorem casesifp 1404
Description: Version of cases 973 expressed using if-. Case disjunction according to the value of  ph. One can see this as a proof that the two hypotheses characterize the conditional operator for propositions. For the converses, see ifptru 1399 and ifpfal 1401. (Contributed by BJ, 20-Sep-2019.)
Hypotheses
Ref Expression
casesifp.1  |-  ( ph  ->  ( ps  <->  ch )
)
casesifp.2  |-  ( -. 
ph  ->  ( ps  <->  th )
)
Assertion
Ref Expression
casesifp  |-  ( ps  <-> if- (
ph ,  ch ,  th ) )

Proof of Theorem casesifp
StepHypRef Expression
1 casesifp.1 . . 3  |-  ( ph  ->  ( ps  <->  ch )
)
2 casesifp.2 . . 3  |-  ( -. 
ph  ->  ( ps  <->  th )
)
31, 2cases 973 . 2  |-  ( ps  <->  ( ( ph  /\  ch )  \/  ( -.  ph 
/\  th ) ) )
4 df-ifp 1389 . 2  |-  (if- (
ph ,  ch ,  th )  <->  ( ( ph  /\ 
ch )  \/  ( -.  ph  /\  th )
) )
53, 4bitr4i 254 1  |-  ( ps  <-> if- (
ph ,  ch ,  th ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 186    \/ wo 368    /\ wa 369  if-wif 1388
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 187  df-or 370  df-an 371  df-ifp 1389
This theorem is referenced by:  cadifp  1489
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