Theorem ifpbi3 44453

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 351	. . 3 ⊢ ((𝜑 ↔ 𝜓) → ((¬ 𝜒 → 𝜑) ↔ (¬ 𝜒 → 𝜓)))
2	1	anbi2d 642	. 2 ⊢ ((𝜑 ↔ 𝜓) → (((𝜒 → 𝜃) ∧ (¬ 𝜒 → 𝜑)) ↔ ((𝜒 → 𝜃) ∧ (¬ 𝜒 → 𝜓))))
3		dfifp2 1080	. 2 ⊢ (if-(𝜒, 𝜃, 𝜑) ↔ ((𝜒 → 𝜃) ∧ (¬ 𝜒 → 𝜑)))
4		dfifp2 1080	. 2 ⊢ (if-(𝜒, 𝜃, 𝜓) ↔ ((𝜒 → 𝜃) ∧ (¬ 𝜒 → 𝜓)))
5	2, 3, 4	3bitr4g 317	1 ⊢ ((𝜑 ↔ 𝜓) → (if-(𝜒, 𝜃, 𝜑) ↔ if-(𝜒, 𝜃, 𝜓)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401  if-wif 1078

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ifp 1079

This theorem is used by:  ifpxorcor  44461  ifpnot23c  44469  ifpdfnan  44471