Theorem ifpbibib 41015

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

Assertion

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

Proof of Theorem ifpbibib

Step	Hyp	Ref	Expression
1		dfifp2 1061	. 2 ⊢ (if-(𝜑, (𝜓 ↔ 𝜒), (𝜃 ↔ 𝜏)) ↔ ((𝜑 → (𝜓 ↔ 𝜒)) ∧ (¬ 𝜑 → (𝜃 ↔ 𝜏))))
2		dfbi2 474	. . . . . 6 ⊢ ((𝜓 ↔ 𝜒) ↔ ((𝜓 → 𝜒) ∧ (𝜒 → 𝜓)))
3	2	imbi2i 335	. . . . 5 ⊢ ((𝜑 → (𝜓 ↔ 𝜒)) ↔ (𝜑 → ((𝜓 → 𝜒) ∧ (𝜒 → 𝜓))))
4		jcab 517	. . . . 5 ⊢ ((𝜑 → ((𝜓 → 𝜒) ∧ (𝜒 → 𝜓))) ↔ ((𝜑 → (𝜓 → 𝜒)) ∧ (𝜑 → (𝜒 → 𝜓))))
5	3, 4	bitri 274	. . . 4 ⊢ ((𝜑 → (𝜓 ↔ 𝜒)) ↔ ((𝜑 → (𝜓 → 𝜒)) ∧ (𝜑 → (𝜒 → 𝜓))))
6		dfbi2 474	. . . . . 6 ⊢ ((𝜃 ↔ 𝜏) ↔ ((𝜃 → 𝜏) ∧ (𝜏 → 𝜃)))
7	6	imbi2i 335	. . . . 5 ⊢ ((¬ 𝜑 → (𝜃 ↔ 𝜏)) ↔ (¬ 𝜑 → ((𝜃 → 𝜏) ∧ (𝜏 → 𝜃))))
8		jcab 517	. . . . 5 ⊢ ((¬ 𝜑 → ((𝜃 → 𝜏) ∧ (𝜏 → 𝜃))) ↔ ((¬ 𝜑 → (𝜃 → 𝜏)) ∧ (¬ 𝜑 → (𝜏 → 𝜃))))
9	7, 8	bitri 274	. . . 4 ⊢ ((¬ 𝜑 → (𝜃 ↔ 𝜏)) ↔ ((¬ 𝜑 → (𝜃 → 𝜏)) ∧ (¬ 𝜑 → (𝜏 → 𝜃))))
10	5, 9	anbi12i 626	. . 3 ⊢ (((𝜑 → (𝜓 ↔ 𝜒)) ∧ (¬ 𝜑 → (𝜃 ↔ 𝜏))) ↔ (((𝜑 → (𝜓 → 𝜒)) ∧ (𝜑 → (𝜒 → 𝜓))) ∧ ((¬ 𝜑 → (𝜃 → 𝜏)) ∧ (¬ 𝜑 → (𝜏 → 𝜃)))))
11		an4 652	. . 3 ⊢ ((((𝜑 → (𝜓 → 𝜒)) ∧ (𝜑 → (𝜒 → 𝜓))) ∧ ((¬ 𝜑 → (𝜃 → 𝜏)) ∧ (¬ 𝜑 → (𝜏 → 𝜃)))) ↔ (((𝜑 → (𝜓 → 𝜒)) ∧ (¬ 𝜑 → (𝜃 → 𝜏))) ∧ ((𝜑 → (𝜒 → 𝜓)) ∧ (¬ 𝜑 → (𝜏 → 𝜃)))))
12	10, 11	bitri 274	. 2 ⊢ (((𝜑 → (𝜓 ↔ 𝜒)) ∧ (¬ 𝜑 → (𝜃 ↔ 𝜏))) ↔ (((𝜑 → (𝜓 → 𝜒)) ∧ (¬ 𝜑 → (𝜃 → 𝜏))) ∧ ((𝜑 → (𝜒 → 𝜓)) ∧ (¬ 𝜑 → (𝜏 → 𝜃)))))
13		dfifp2 1061	. . . . 5 ⊢ (if-(𝜑, (𝜓 → 𝜒), (𝜃 → 𝜏)) ↔ ((𝜑 → (𝜓 → 𝜒)) ∧ (¬ 𝜑 → (𝜃 → 𝜏))))
14		ifpimimb 41009	. . . . 5 ⊢ (if-(𝜑, (𝜓 → 𝜒), (𝜃 → 𝜏)) ↔ (if-(𝜑, 𝜓, 𝜃) → if-(𝜑, 𝜒, 𝜏)))
15	13, 14	bitr3i 276	. . . 4 ⊢ (((𝜑 → (𝜓 → 𝜒)) ∧ (¬ 𝜑 → (𝜃 → 𝜏))) ↔ (if-(𝜑, 𝜓, 𝜃) → if-(𝜑, 𝜒, 𝜏)))
16		dfifp2 1061	. . . . 5 ⊢ (if-(𝜑, (𝜒 → 𝜓), (𝜏 → 𝜃)) ↔ ((𝜑 → (𝜒 → 𝜓)) ∧ (¬ 𝜑 → (𝜏 → 𝜃))))
17		ifpimimb 41009	. . . . 5 ⊢ (if-(𝜑, (𝜒 → 𝜓), (𝜏 → 𝜃)) ↔ (if-(𝜑, 𝜒, 𝜏) → if-(𝜑, 𝜓, 𝜃)))
18	16, 17	bitr3i 276	. . . 4 ⊢ (((𝜑 → (𝜒 → 𝜓)) ∧ (¬ 𝜑 → (𝜏 → 𝜃))) ↔ (if-(𝜑, 𝜒, 𝜏) → if-(𝜑, 𝜓, 𝜃)))
19	15, 18	anbi12i 626	. . 3 ⊢ ((((𝜑 → (𝜓 → 𝜒)) ∧ (¬ 𝜑 → (𝜃 → 𝜏))) ∧ ((𝜑 → (𝜒 → 𝜓)) ∧ (¬ 𝜑 → (𝜏 → 𝜃)))) ↔ ((if-(𝜑, 𝜓, 𝜃) → if-(𝜑, 𝜒, 𝜏)) ∧ (if-(𝜑, 𝜒, 𝜏) → if-(𝜑, 𝜓, 𝜃))))
20		dfbi2 474	. . 3 ⊢ ((if-(𝜑, 𝜓, 𝜃) ↔ if-(𝜑, 𝜒, 𝜏)) ↔ ((if-(𝜑, 𝜓, 𝜃) → if-(𝜑, 𝜒, 𝜏)) ∧ (if-(𝜑, 𝜒, 𝜏) → if-(𝜑, 𝜓, 𝜃))))
21	19, 20	bitr4i 277	. 2 ⊢ ((((𝜑 → (𝜓 → 𝜒)) ∧ (¬ 𝜑 → (𝜃 → 𝜏))) ∧ ((𝜑 → (𝜒 → 𝜓)) ∧ (¬ 𝜑 → (𝜏 → 𝜃)))) ↔ (if-(𝜑, 𝜓, 𝜃) ↔ if-(𝜑, 𝜒, 𝜏)))
22	1, 12, 21	3bitri 296	1 ⊢ (if-(𝜑, (𝜓 ↔ 𝜒), (𝜃 ↔ 𝜏)) ↔ (if-(𝜑, 𝜓, 𝜃) ↔ if-(𝜑, 𝜒, 𝜏)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395  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:  ifpxorxorb  41016