Theorem ifpbi23

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

	Ref	Expression
Assertion	ifpbi23	⊢ ( ( ( 𝜑 ↔ 𝜓 ) ∧ ( 𝜒 ↔ 𝜃 ) ) → ( if- ( 𝜏 , 𝜑 , 𝜒 ) ↔ if- ( 𝜏 , 𝜓 , 𝜃 ) ) )

Proof

Step	Hyp	Ref	Expression
1		simpl	⊢ ( ( ( 𝜑 ↔ 𝜓 ) ∧ ( 𝜒 ↔ 𝜃 ) ) → ( 𝜑 ↔ 𝜓 ) )
2		simpr	⊢ ( ( ( 𝜑 ↔ 𝜓 ) ∧ ( 𝜒 ↔ 𝜃 ) ) → ( 𝜒 ↔ 𝜃 ) )
3	1 2	ifpbi23d	⊢ ( ( ( 𝜑 ↔ 𝜓 ) ∧ ( 𝜒 ↔ 𝜃 ) ) → ( if- ( 𝜏 , 𝜑 , 𝜒 ) ↔ if- ( 𝜏 , 𝜓 , 𝜃 ) ) )