Theorem ifpbi3 37812

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

Assertion

Ref	Expression
ifpbi3	⊢ ((𝜑 ↔ 𝜓) → (if-(𝜒, 𝜃, 𝜑) ↔ if-(𝜒, 𝜃, 𝜓)))

Proof of Theorem ifpbi3

Step	Hyp	Ref	Expression
1		imbi2 338	. . 3 ⊢ ((𝜑 ↔ 𝜓) → ((¬ 𝜒 → 𝜑) ↔ (¬ 𝜒 → 𝜓)))
2	1	anbi2d 740	. 2 ⊢ ((𝜑 ↔ 𝜓) → (((𝜒 → 𝜃) ∧ (¬ 𝜒 → 𝜑)) ↔ ((𝜒 → 𝜃) ∧ (¬ 𝜒 → 𝜓))))
3		dfifp2 1014	. 2 ⊢ (if-(𝜒, 𝜃, 𝜑) ↔ ((𝜒 → 𝜃) ∧ (¬ 𝜒 → 𝜑)))
4		dfifp2 1014	. 2 ⊢ (if-(𝜒, 𝜃, 𝜓) ↔ ((𝜒 → 𝜃) ∧ (¬ 𝜒 → 𝜓)))
5	2, 3, 4	3bitr4g 303	1 ⊢ ((𝜑 ↔ 𝜓) → (if-(𝜒, 𝜃, 𝜑) ↔ if-(𝜒, 𝜃, 𝜓)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384  if-wif 1012

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 197  df-or 385  df-an 386  df-ifp 1013

This theorem is referenced by:  ifpxorcor  37821  ifpnot23c  37829  ifpdfnan  37831