Theorem dfifp2 1059

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 1058). (Contributed by BJ, 22-Jun-2019.)

Assertion

Ref	Expression
dfifp2	⊢ (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑 → 𝜓) ∧ (¬ 𝜑 → 𝜒)))

Proof of Theorem dfifp2

Step	Hyp	Ref	Expression
1		df-ifp 1058	. 2 ⊢ (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ 𝜒)))
2		cases2 1042	. 2 ⊢ (((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ 𝜒)) ↔ ((𝜑 → 𝜓) ∧ (¬ 𝜑 → 𝜒)))
3	1, 2	bitri 277	1 ⊢ (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑 → 𝜓) ∧ (¬ 𝜑 → 𝜒)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   ∨ wo 843  if-wif 1057

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 209  df-an 399  df-or 844  df-ifp 1058

This theorem is referenced by:  dfifp3  1060  dfifp5  1062  ifpn  1067  ifpimpda  1074  revwlk  32371  wl-ifpdfbi  34749  ifpbi2  39852  ifpbi3  39853  ifpbi23  39858  ifpbi1  39863  ifpbi12  39874  ifpbi13  39875  ifpimimb  39890  ifpororb  39891  ifpbibib  39896  frege54cor0a  40229