Theorem ifpananb 41075

Description: Factor conditional logic operator over conjunction in terms 2 and 3. (Contributed by RP, 21-Apr-2020.)

Assertion

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

Proof of Theorem ifpananb

Step	Hyp	Ref	Expression
1		anor 979	. . 3 ⊢ ((𝜓 ∧ 𝜒) ↔ ¬ (¬ 𝜓 ∨ ¬ 𝜒))
2		anor 979	. . 3 ⊢ ((𝜃 ∧ 𝜏) ↔ ¬ (¬ 𝜃 ∨ ¬ 𝜏))
3		ifpbi23 41042	. . 3 ⊢ ((((𝜓 ∧ 𝜒) ↔ ¬ (¬ 𝜓 ∨ ¬ 𝜒)) ∧ ((𝜃 ∧ 𝜏) ↔ ¬ (¬ 𝜃 ∨ ¬ 𝜏))) → (if-(𝜑, (𝜓 ∧ 𝜒), (𝜃 ∧ 𝜏)) ↔ if-(𝜑, ¬ (¬ 𝜓 ∨ ¬ 𝜒), ¬ (¬ 𝜃 ∨ ¬ 𝜏))))
4	1, 2, 3	mp2an 688	. 2 ⊢ (if-(𝜑, (𝜓 ∧ 𝜒), (𝜃 ∧ 𝜏)) ↔ if-(𝜑, ¬ (¬ 𝜓 ∨ ¬ 𝜒), ¬ (¬ 𝜃 ∨ ¬ 𝜏)))
5		ifpororb 41074	. . . . 5 ⊢ (if-(𝜑, (¬ 𝜓 ∨ ¬ 𝜒), (¬ 𝜃 ∨ ¬ 𝜏)) ↔ (if-(𝜑, ¬ 𝜓, ¬ 𝜃) ∨ if-(𝜑, ¬ 𝜒, ¬ 𝜏)))
6		ifpnotnotb 41048	. . . . . 6 ⊢ (if-(𝜑, ¬ 𝜓, ¬ 𝜃) ↔ ¬ if-(𝜑, 𝜓, 𝜃))
7		ifpnotnotb 41048	. . . . . 6 ⊢ (if-(𝜑, ¬ 𝜒, ¬ 𝜏) ↔ ¬ if-(𝜑, 𝜒, 𝜏))
8	6, 7	orbi12i 911	. . . . 5 ⊢ ((if-(𝜑, ¬ 𝜓, ¬ 𝜃) ∨ if-(𝜑, ¬ 𝜒, ¬ 𝜏)) ↔ (¬ if-(𝜑, 𝜓, 𝜃) ∨ ¬ if-(𝜑, 𝜒, 𝜏)))
9	5, 8	bitri 274	. . . 4 ⊢ (if-(𝜑, (¬ 𝜓 ∨ ¬ 𝜒), (¬ 𝜃 ∨ ¬ 𝜏)) ↔ (¬ if-(𝜑, 𝜓, 𝜃) ∨ ¬ if-(𝜑, 𝜒, 𝜏)))
10	9	notbii 319	. . 3 ⊢ (¬ if-(𝜑, (¬ 𝜓 ∨ ¬ 𝜒), (¬ 𝜃 ∨ ¬ 𝜏)) ↔ ¬ (¬ if-(𝜑, 𝜓, 𝜃) ∨ ¬ if-(𝜑, 𝜒, 𝜏)))
11		ifpnotnotb 41048	. . 3 ⊢ (if-(𝜑, ¬ (¬ 𝜓 ∨ ¬ 𝜒), ¬ (¬ 𝜃 ∨ ¬ 𝜏)) ↔ ¬ if-(𝜑, (¬ 𝜓 ∨ ¬ 𝜒), (¬ 𝜃 ∨ ¬ 𝜏)))
12		anor 979	. . 3 ⊢ ((if-(𝜑, 𝜓, 𝜃) ∧ if-(𝜑, 𝜒, 𝜏)) ↔ ¬ (¬ if-(𝜑, 𝜓, 𝜃) ∨ ¬ if-(𝜑, 𝜒, 𝜏)))
13	10, 11, 12	3bitr4i 302	. 2 ⊢ (if-(𝜑, ¬ (¬ 𝜓 ∨ ¬ 𝜒), ¬ (¬ 𝜃 ∨ ¬ 𝜏)) ↔ (if-(𝜑, 𝜓, 𝜃) ∧ if-(𝜑, 𝜒, 𝜏)))
14	4, 13	bitri 274	1 ⊢ (if-(𝜑, (𝜓 ∧ 𝜒), (𝜃 ∧ 𝜏)) ↔ (if-(𝜑, 𝜓, 𝜃) ∧ if-(𝜑, 𝜒, 𝜏)))

Syntax hints:  ¬ wn 3   ↔ wb 205   ∧ wa 395   ∨ wo 843  if-wif 1059

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 206  df-an 396  df-or 844  df-ifp 1060

This theorem is referenced by:  ifpnannanb  41076