Theorem ifpan23 44246

Description: Conjunction of conditional logical operators. (Contributed by RP, 20-Apr-2020.)

Assertion

Ref	Expression
ifpan23	⊢ ((if-(𝜑, 𝜓, 𝜒) ∧ if-(𝜑, 𝜃, 𝜏)) ↔ if-(𝜑, (𝜓 ∧ 𝜃), (𝜒 ∧ 𝜏)))

Proof of Theorem ifpan23

Step	Hyp	Ref	Expression
1		ifpan123g 44245	. 2 ⊢ ((if-(𝜑, 𝜓, 𝜒) ∧ if-(𝜑, 𝜃, 𝜏)) ↔ (((¬ 𝜑 ∨ 𝜓) ∧ (𝜑 ∨ 𝜒)) ∧ ((¬ 𝜑 ∨ 𝜃) ∧ (𝜑 ∨ 𝜏))))
2		an4 669	. 2 ⊢ ((((¬ 𝜑 ∨ 𝜓) ∧ (𝜑 ∨ 𝜒)) ∧ ((¬ 𝜑 ∨ 𝜃) ∧ (𝜑 ∨ 𝜏))) ↔ (((¬ 𝜑 ∨ 𝜓) ∧ (¬ 𝜑 ∨ 𝜃)) ∧ ((𝜑 ∨ 𝜒) ∧ (𝜑 ∨ 𝜏))))
3		dfifp4 1082	. . 3 ⊢ (if-(𝜑, (𝜓 ∧ 𝜃), (𝜒 ∧ 𝜏)) ↔ ((¬ 𝜑 ∨ (𝜓 ∧ 𝜃)) ∧ (𝜑 ∨ (𝜒 ∧ 𝜏))))
4		ordi 1023	. . . 4 ⊢ ((¬ 𝜑 ∨ (𝜓 ∧ 𝜃)) ↔ ((¬ 𝜑 ∨ 𝜓) ∧ (¬ 𝜑 ∨ 𝜃)))
5		ordi 1023	. . . 4 ⊢ ((𝜑 ∨ (𝜒 ∧ 𝜏)) ↔ ((𝜑 ∨ 𝜒) ∧ (𝜑 ∨ 𝜏)))
6	4, 5	anbi12i 640	. . 3 ⊢ (((¬ 𝜑 ∨ (𝜓 ∧ 𝜃)) ∧ (𝜑 ∨ (𝜒 ∧ 𝜏))) ↔ (((¬ 𝜑 ∨ 𝜓) ∧ (¬ 𝜑 ∨ 𝜃)) ∧ ((𝜑 ∨ 𝜒) ∧ (𝜑 ∨ 𝜏))))
7	3, 6	bitr2i 279	. 2 ⊢ ((((¬ 𝜑 ∨ 𝜓) ∧ (¬ 𝜑 ∨ 𝜃)) ∧ ((𝜑 ∨ 𝜒) ∧ (𝜑 ∨ 𝜏))) ↔ if-(𝜑, (𝜓 ∧ 𝜃), (𝜒 ∧ 𝜏)))
8	1, 2, 7	3bitri 300	1 ⊢ ((if-(𝜑, 𝜓, 𝜒) ∧ if-(𝜑, 𝜃, 𝜏)) ↔ if-(𝜑, (𝜓 ∧ 𝜃), (𝜒 ∧ 𝜏)))

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209   ∧ wa 401   ∨ wo 861  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 402  df-or 862  df-ifp 1079

This theorem is used by:  ifpdfxor  44273