Theorem ifpbi123dOLD 1078

Description: Obsolete version of ifpbi123d 1077 as of 17-Apr-2024. (Contributed by AV, 30-Dec-2020.) (New usage is discouraged.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
ifpbi123d.1	⊢ (𝜑 → (𝜓 ↔ 𝜏))
ifpbi123d.2	⊢ (𝜑 → (𝜒 ↔ 𝜂))
ifpbi123d.3	⊢ (𝜑 → (𝜃 ↔ 𝜁))

Assertion

Ref	Expression
ifpbi123dOLD	⊢ (𝜑 → (if-(𝜓, 𝜒, 𝜃) ↔ if-(𝜏, 𝜂, 𝜁)))

Proof of Theorem ifpbi123dOLD

Step	Hyp	Ref	Expression
1		ifpbi123d.1	. . . 4 ⊢ (𝜑 → (𝜓 ↔ 𝜏))
2		ifpbi123d.2	. . . 4 ⊢ (𝜑 → (𝜒 ↔ 𝜂))
3	1, 2	anbi12d 631	. . 3 ⊢ (𝜑 → ((𝜓 ∧ 𝜒) ↔ (𝜏 ∧ 𝜂)))
4	1	notbid 318	. . . 4 ⊢ (𝜑 → (¬ 𝜓 ↔ ¬ 𝜏))
5		ifpbi123d.3	. . . 4 ⊢ (𝜑 → (𝜃 ↔ 𝜁))
6	4, 5	anbi12d 631	. . 3 ⊢ (𝜑 → ((¬ 𝜓 ∧ 𝜃) ↔ (¬ 𝜏 ∧ 𝜁)))
7	3, 6	orbi12d 916	. 2 ⊢ (𝜑 → (((𝜓 ∧ 𝜒) ∨ (¬ 𝜓 ∧ 𝜃)) ↔ ((𝜏 ∧ 𝜂) ∨ (¬ 𝜏 ∧ 𝜁))))
8		df-ifp 1061	. 2 ⊢ (if-(𝜓, 𝜒, 𝜃) ↔ ((𝜓 ∧ 𝜒) ∨ (¬ 𝜓 ∧ 𝜃)))
9		df-ifp 1061	. 2 ⊢ (if-(𝜏, 𝜂, 𝜁) ↔ ((𝜏 ∧ 𝜂) ∨ (¬ 𝜏 ∧ 𝜁)))
10	7, 8, 9	3bitr4g 314	1 ⊢ (𝜑 → (if-(𝜓, 𝜒, 𝜃) ↔ if-(𝜏, 𝜂, 𝜁)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 844  if-wif 1060

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 397  df-or 845  df-ifp 1061

This theorem is referenced by: (None)