MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dfifp2 Structured version   Visualization version   GIF version

Theorem dfifp2 1080
Description: Alternate definition of the conditional operator for propositions. The value of if-(𝜑, 𝜓, 𝜒) is "if 𝜑 then 𝜓, and if not 𝜑 then 𝜒". This is the definition used in Section II.24 of [Church] p. 129 (Definition D12 page 132) (see comment of df-ifp 1079). (Contributed by BJ, 22-Jun-2019.)
Assertion
Ref Expression
dfifp2 (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑𝜓) ∧ (¬ 𝜑𝜒)))

Proof of Theorem dfifp2
StepHypRef Expression
1 df-ifp 1079 . 2 (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑𝜓) ∨ (¬ 𝜑𝜒)))
2 cases2 1061 . 2 (((𝜑𝜓) ∨ (¬ 𝜑𝜒)) ↔ ((𝜑𝜓) ∧ (¬ 𝜑𝜒)))
31, 2bitri 266 1 (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑𝜓) ∧ (¬ 𝜑𝜒)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  wo 865  if-wif 1078
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 198  df-an 385  df-or 866  df-ifp 1079
This theorem is referenced by:  dfifp3  1081  dfifp5  1083  ifpn  1088  ifpimpda  1094  ifpbi2  38305  ifpbi3  38306  ifpbi23  38311  ifpbi1  38316  ifpbi12  38327  ifpbi13  38328  ifpbi123  38329  ifpimimb  38343  ifpororb  38344  ifpbibib  38349  frege54cor0a  38651
  Copyright terms: Public domain W3C validator