Theorem dfifp2 1061

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 1060). (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 1060	. 2 ⊢ (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ 𝜒)))
2		cases2 1044	. 2 ⊢ (((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ 𝜒)) ↔ ((𝜑 → 𝜓) ∧ (¬ 𝜑 → 𝜒)))
3	1, 2	bitri 274	1 ⊢ (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑 → 𝜓) ∧ (¬ 𝜑 → 𝜒)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 394   ∨ wo 843  if-wif 1059

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 206  df-an 395  df-or 844  df-ifp 1060

This theorem is referenced by:  dfifp3  1062  dfifp5  1064  ifpdfbi  1067  ifpnOLD  1071  ifpimpda  1079  revwlk  34413  ifpbi2  42520  ifpbi3  42521  ifpbi1  42530  ifpbi12  42541  ifpbi13  42542  ifpimimb  42557  ifpororb  42558  ifpbibib  42563  frege54cor0a  42916