Theorem ifpbi123

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

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

Proof

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