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Theorem dfifp2 1034
 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 1033). (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 1033 . 2 (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑𝜓) ∨ (¬ 𝜑𝜒)))
2 cases2 1016 . 2 (((𝜑𝜓) ∨ (¬ 𝜑𝜒)) ↔ ((𝜑𝜓) ∧ (¬ 𝜑𝜒)))
31, 2bitri 264 1 (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑𝜓) ∧ (¬ 𝜑𝜒)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 382   ∧ wa 383  if-wif 1032 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 197  df-or 384  df-an 385  df-ifp 1033 This theorem is referenced by:  dfifp3  1035  dfifp5  1037  ifpn  1042  ifpimpda  1048  ifpbi2  38128  ifpbi3  38129  ifpbi23  38134  ifpbi1  38139  ifpbi12  38150  ifpbi13  38151  ifpbi123  38152  ifpimimb  38166  ifpororb  38167  ifpbibib  38172  frege54cor0a  38474
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