Theorem dfifp2 1065

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 1064). (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 1064	. 2 ⊢ (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ 𝜒)))
2		cases2 1048	. 2 ⊢ (((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ 𝜒)) ↔ ((𝜑 → 𝜓) ∧ (¬ 𝜑 → 𝜒)))
3	1, 2	bitri 275	1 ⊢ (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑 → 𝜓) ∧ (¬ 𝜑 → 𝜒)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 846  if-wif 1063

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 207  df-an 396  df-or 847  df-ifp 1064

This theorem is referenced by:  dfifp3  1066  dfifp5  1068  ifpdfbi  1071  ifpimpda  1081  revwlk  35092  ifpbi2  43429  ifpbi3  43430  ifpbi1  43439  ifpbi12  43450  ifpbi13  43451  ifpimimb  43466  ifpororb  43467  ifpbibib  43472  frege54cor0a  43825