Theorem dfifp3 1081

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 1080	. 2 ⊢ (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑 → 𝜓) ∧ (¬ 𝜑 → 𝜒)))
2		pm4.64 862	. . 3 ⊢ ((¬ 𝜑 → 𝜒) ↔ (𝜑 ∨ 𝜒))
3	2	anbi2i 634	. 2 ⊢ (((𝜑 → 𝜓) ∧ (¬ 𝜑 → 𝜒)) ↔ ((𝜑 → 𝜓) ∧ (𝜑 ∨ 𝜒)))
4	1, 3	bitri 278	1 ⊢ (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑 → 𝜓) ∧ (𝜑 ∨ 𝜒)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860  if-wif 1078

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-ifp 1079

This theorem is used by:  dfifp4  1082  ifpn  1090  ifptru  1091  ifpbi123d  1095  wl-2mintru1  38164