Theorem ifpbi2 37811

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

Assertion

Ref	Expression
ifpbi2	⊢ ((𝜑 ↔ 𝜓) → (if-(𝜒, 𝜑, 𝜃) ↔ if-(𝜒, 𝜓, 𝜃)))

Proof of Theorem ifpbi2

Step	Hyp	Ref	Expression
1		imbi2 338	. . 3 ⊢ ((𝜑 ↔ 𝜓) → ((𝜒 → 𝜑) ↔ (𝜒 → 𝜓)))
2	1	anbi1d 741	. 2 ⊢ ((𝜑 ↔ 𝜓) → (((𝜒 → 𝜑) ∧ (¬ 𝜒 → 𝜃)) ↔ ((𝜒 → 𝜓) ∧ (¬ 𝜒 → 𝜃))))
3		dfifp2 1014	. 2 ⊢ (if-(𝜒, 𝜑, 𝜃) ↔ ((𝜒 → 𝜑) ∧ (¬ 𝜒 → 𝜃)))
4		dfifp2 1014	. 2 ⊢ (if-(𝜒, 𝜓, 𝜃) ↔ ((𝜒 → 𝜓) ∧ (¬ 𝜒 → 𝜃)))
5	2, 3, 4	3bitr4g 303	1 ⊢ ((𝜑 ↔ 𝜓) → (if-(𝜒, 𝜑, 𝜃) ↔ if-(𝜒, 𝜓, 𝜃)))

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:  ifpnot23b  37827