Theorem dfifp3 1046

Description: Alternate definition of the conditional operator for propositions. (Contributed by BJ, 30-Sep-2019.)

Assertion

Ref	Expression
dfifp3	⊢ (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑 → 𝜓) ∧ (𝜑 ∨ 𝜒)))

Proof of Theorem dfifp3

Step	Hyp	Ref	Expression
1		dfifp2 1045	. 2 ⊢ (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑 → 𝜓) ∧ (¬ 𝜑 → 𝜒)))
2		pm4.64 835	. . 3 ⊢ ((¬ 𝜑 → 𝜒) ↔ (𝜑 ∨ 𝜒))
3	2	anbi2i 613	. 2 ⊢ (((𝜑 → 𝜓) ∧ (¬ 𝜑 → 𝜒)) ↔ ((𝜑 → 𝜓) ∧ (𝜑 ∨ 𝜒)))
4	1, 3	bitri 267	1 ⊢ (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑 → 𝜓) ∧ (𝜑 ∨ 𝜒)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 198   ∧ wa 387   ∨ wo 833  if-wif 1043

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 199  df-an 388  df-or 834  df-ifp 1044

This theorem is referenced by:  dfifp4  1047  ifptru  1054