Theorem dfifp4 1067

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

Assertion

Ref	Expression
dfifp4	⊢ (if-(𝜑, 𝜓, 𝜒) ↔ ((¬ 𝜑 ∨ 𝜓) ∧ (𝜑 ∨ 𝜒)))

Proof of Theorem dfifp4

Step	Hyp	Ref	Expression
1		dfifp3 1066	. 2 ⊢ (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑 → 𝜓) ∧ (𝜑 ∨ 𝜒)))
2		imor 854	. 2 ⊢ ((𝜑 → 𝜓) ↔ (¬ 𝜑 ∨ 𝜓))
3	1, 2	bianbi 628	1 ⊢ (if-(𝜑, 𝜓, 𝜒) ↔ ((¬ 𝜑 ∨ 𝜓) ∧ (𝜑 ∨ 𝜒)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848  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 849  df-ifp 1064

This theorem is referenced by:  anifp  1072  ifpan123g  43908  ifpan23  43909  ifpdfor2  43910  ifpdfor  43914  ifpim1  43918  ifpnot  43919  ifpid2  43920  ifpim2  43921  ifpnot23  43927  ifpidg  43940  ifpim123g  43949  ifpimim  43958