Theorem ifpbi123 44257

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

Assertion

Ref	Expression
ifpbi123	⊢ (((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃) ∧ (𝜏 ↔ 𝜂)) → (if-(𝜑, 𝜒, 𝜏) ↔ if-(𝜓, 𝜃, 𝜂)))

Proof of Theorem ifpbi123

Step	Hyp	Ref	Expression
1		simp1 1154	. 2 ⊢ (((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃) ∧ (𝜏 ↔ 𝜂)) → (𝜑 ↔ 𝜓))
2		simp2 1155	. 2 ⊢ (((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃) ∧ (𝜏 ↔ 𝜂)) → (𝜒 ↔ 𝜃))
3		simp3 1156	. 2 ⊢ (((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃) ∧ (𝜏 ↔ 𝜂)) → (𝜏 ↔ 𝜂))
4	1, 2, 3	ifpbi123d 1095	1 ⊢ (((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃) ∧ (𝜏 ↔ 𝜂)) → (if-(𝜑, 𝜒, 𝜏) ↔ if-(𝜓, 𝜃, 𝜂)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  if-wif 1078   ∧ w3a 1103

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  df-3an 1105

This theorem is used by: (None)