MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dfifp3 Structured version   Visualization version   Unicode version

Theorem dfifp3 1427
Description: Alternate definition of the conditional operator for propositions. (Contributed by BJ, 30-Sep-2019.)
Assertion
Ref Expression
dfifp3  |-  (if- (
ph ,  ps ,  ch )  <->  ( ( ph  ->  ps )  /\  ( ph  \/  ch ) ) )

Proof of Theorem dfifp3
StepHypRef Expression
1 dfifp2 1426 . 2  |-  (if- (
ph ,  ps ,  ch )  <->  ( ( ph  ->  ps )  /\  ( -.  ph  ->  ch )
) )
2 pm4.64 374 . . 3  |-  ( ( -.  ph  ->  ch )  <->  (
ph  \/  ch )
)
32anbi2i 699 . 2  |-  ( ( ( ph  ->  ps )  /\  ( -.  ph  ->  ch ) )  <->  ( ( ph  ->  ps )  /\  ( ph  \/  ch )
) )
41, 3bitri 253 1  |-  (if- (
ph ,  ps ,  ch )  <->  ( ( ph  ->  ps )  /\  ( ph  \/  ch ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 188    \/ wo 370    /\ wa 371  if-wif 1424
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 1425
This theorem is referenced by:  dfifp4  1428  ifptru  1435
  Copyright terms: Public domain W3C validator