Theorem ifpan123g 43472

Description: Conjunction of conditional logical operators. (Contributed by RP, 18-Apr-2020.)

Assertion

Ref	Expression
ifpan123g	⊢ ((if-(𝜑, 𝜒, 𝜏) ∧ if-(𝜓, 𝜃, 𝜂)) ↔ (((¬ 𝜑 ∨ 𝜒) ∧ (𝜑 ∨ 𝜏)) ∧ ((¬ 𝜓 ∨ 𝜃) ∧ (𝜓 ∨ 𝜂))))

Proof of Theorem ifpan123g

Step	Hyp	Ref	Expression
1		dfifp4 1067	. 2 ⊢ (if-(𝜑, 𝜒, 𝜏) ↔ ((¬ 𝜑 ∨ 𝜒) ∧ (𝜑 ∨ 𝜏)))
2		dfifp4 1067	. 2 ⊢ (if-(𝜓, 𝜃, 𝜂) ↔ ((¬ 𝜓 ∨ 𝜃) ∧ (𝜓 ∨ 𝜂)))
3	1, 2	anbi12i 628	1 ⊢ ((if-(𝜑, 𝜒, 𝜏) ∧ if-(𝜓, 𝜃, 𝜂)) ↔ (((¬ 𝜑 ∨ 𝜒) ∧ (𝜑 ∨ 𝜏)) ∧ ((¬ 𝜓 ∨ 𝜃) ∧ (𝜓 ∨ 𝜂))))

Syntax hints:  ¬ wn 3   ↔ 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:  ifpan23  43473