Theorem ifpbi12 40993

Description: Equivalence theorem for conditional logical operators. (Contributed by RP, 15-Apr-2020.)

Assertion

Ref	Expression
ifpbi12	⊢ (((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃)) → (if-(𝜑, 𝜒, 𝜏) ↔ if-(𝜓, 𝜃, 𝜏)))

Proof of Theorem ifpbi12

Step	Hyp	Ref	Expression
1		imbi12 346	. . . 4 ⊢ ((𝜑 ↔ 𝜓) → ((𝜒 ↔ 𝜃) → ((𝜑 → 𝜒) ↔ (𝜓 → 𝜃))))
2	1	imp 406	. . 3 ⊢ (((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃)) → ((𝜑 → 𝜒) ↔ (𝜓 → 𝜃)))
3		simpl 482	. . . . 5 ⊢ (((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃)) → (𝜑 ↔ 𝜓))
4	3	notbid 317	. . . 4 ⊢ (((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃)) → (¬ 𝜑 ↔ ¬ 𝜓))
5	4	imbi1d 341	. . 3 ⊢ (((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃)) → ((¬ 𝜑 → 𝜏) ↔ (¬ 𝜓 → 𝜏)))
6	2, 5	anbi12d 630	. 2 ⊢ (((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃)) → (((𝜑 → 𝜒) ∧ (¬ 𝜑 → 𝜏)) ↔ ((𝜓 → 𝜃) ∧ (¬ 𝜓 → 𝜏))))
7		dfifp2 1061	. 2 ⊢ (if-(𝜑, 𝜒, 𝜏) ↔ ((𝜑 → 𝜒) ∧ (¬ 𝜑 → 𝜏)))
8		dfifp2 1061	. 2 ⊢ (if-(𝜓, 𝜃, 𝜏) ↔ ((𝜓 → 𝜃) ∧ (¬ 𝜓 → 𝜏)))
9	6, 7, 8	3bitr4g 313	1 ⊢ (((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃)) → (if-(𝜑, 𝜒, 𝜏) ↔ if-(𝜓, 𝜃, 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: (None)