Theorem dfifp2 1057

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 version of the definition uses only primitive symbols (→ , ¬ , ∀). This is the definition used in Section II.24 of [Church] p. 129 (Definition D12 page 132) (see comment of df-ifp 1056). (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 1056	. 2 ⊢ (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ 𝜒)))
2		cases2 1040	. 2 ⊢ (((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ 𝜒)) ↔ ((𝜑 → 𝜓) ∧ (¬ 𝜑 → 𝜒)))
3	1, 2	bitri 276	1 ⊢ (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑 → 𝜓) ∧ (¬ 𝜑 → 𝜒)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396   ∨ wo 842  if-wif 1055

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 208  df-an 397  df-or 843  df-ifp 1056

This theorem is referenced by:  dfifp3  1058  dfifp5  1060  ifpn  1065  ifpimpda  1071  revwlk  31987  ifpbi2  39342  ifpbi3  39343  ifpbi23  39348  ifpbi1  39353  ifpbi12  39364  ifpbi13  39365  ifpimimb  39380  ifpororb  39381  ifpbibib  39386  frege54cor0a  39719