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 853	. . 3 ⊢ ((𝜑 → 𝜓) ↔ (¬ 𝜑 ∨ 𝜓))
3	2	anbi1i 627	. 2 ⊢ (((𝜑 → 𝜓) ∧ (𝜑 ∨ 𝜒)) ↔ ((¬ 𝜑 ∨ 𝜓) ∧ (𝜑 ∨ 𝜒)))
4	1, 3	bitri 278	1 ⊢ (if-(𝜑, 𝜓, 𝜒) ↔ ((¬ 𝜑 ∨ 𝜓) ∧ (𝜑 ∨ 𝜒)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 847  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 210  df-an 400  df-or 848  df-ifp 1064

This theorem is referenced by:  anifp  1072  ifpan123g  40692  ifpan23  40693  ifpdfor2  40694  ifpdfor  40698  ifpim1  40702  ifpnot  40703  ifpid2  40704  ifpim2  40705  ifpnot23  40711  ifpidg  40724  ifpim123g  40733  ifpimim  40742