Theorem ifpbi1 42693

Description: Equivalence theorem for conditional logical operators. (Contributed by RP, 14-Apr-2020.)

Assertion

Ref	Expression
ifpbi1	⊢ ((𝜑 ↔ 𝜓) → (if-(𝜑, 𝜒, 𝜃) ↔ if-(𝜓, 𝜒, 𝜃)))

Proof of Theorem ifpbi1

Step	Hyp	Ref	Expression
1		imbi1 347	. . 3 ⊢ ((𝜑 ↔ 𝜓) → ((𝜑 → 𝜒) ↔ (𝜓 → 𝜒)))
2		notbi 319	. . . . 5 ⊢ ((𝜑 ↔ 𝜓) ↔ (¬ 𝜑 ↔ ¬ 𝜓))
3	2	biimpi 215	. . . 4 ⊢ ((𝜑 ↔ 𝜓) → (¬ 𝜑 ↔ ¬ 𝜓))
4	3	imbi1d 341	. . 3 ⊢ ((𝜑 ↔ 𝜓) → ((¬ 𝜑 → 𝜃) ↔ (¬ 𝜓 → 𝜃)))
5	1, 4	anbi12d 630	. 2 ⊢ ((𝜑 ↔ 𝜓) → (((𝜑 → 𝜒) ∧ (¬ 𝜑 → 𝜃)) ↔ ((𝜓 → 𝜒) ∧ (¬ 𝜓 → 𝜃))))
6		dfifp2 1062	. 2 ⊢ (if-(𝜑, 𝜒, 𝜃) ↔ ((𝜑 → 𝜒) ∧ (¬ 𝜑 → 𝜃)))
7		dfifp2 1062	. 2 ⊢ (if-(𝜓, 𝜒, 𝜃) ↔ ((𝜓 → 𝜒) ∧ (¬ 𝜓 → 𝜃)))
8	5, 6, 7	3bitr4g 314	1 ⊢ ((𝜑 ↔ 𝜓) → (if-(𝜑, 𝜒, 𝜃) ↔ if-(𝜓, 𝜒, 𝜃)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395  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 396  df-or 845  df-ifp 1061

This theorem is referenced by:  ifpimim  42725  axfrege52a  43072