Theorem ifpbi3

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

	Ref	Expression
Assertion	ifpbi3	⊢ ( ( 𝜑 ↔ 𝜓 ) → ( if- ( 𝜒 , 𝜃 , 𝜑 ) ↔ if- ( 𝜒 , 𝜃 , 𝜓 ) ) )

Proof

Step	Hyp	Ref	Expression
1		imbi2	⊢ ( ( 𝜑 ↔ 𝜓 ) → ( ( ¬ 𝜒 → 𝜑 ) ↔ ( ¬ 𝜒 → 𝜓 ) ) )
2	1	anbi2d	⊢ ( ( 𝜑 ↔ 𝜓 ) → ( ( ( 𝜒 → 𝜃 ) ∧ ( ¬ 𝜒 → 𝜑 ) ) ↔ ( ( 𝜒 → 𝜃 ) ∧ ( ¬ 𝜒 → 𝜓 ) ) ) )
3		dfifp2	⊢ ( if- ( 𝜒 , 𝜃 , 𝜑 ) ↔ ( ( 𝜒 → 𝜃 ) ∧ ( ¬ 𝜒 → 𝜑 ) ) )
4		dfifp2	⊢ ( if- ( 𝜒 , 𝜃 , 𝜓 ) ↔ ( ( 𝜒 → 𝜃 ) ∧ ( ¬ 𝜒 → 𝜓 ) ) )
5	2 3 4	3bitr4g	⊢ ( ( 𝜑 ↔ 𝜓 ) → ( if- ( 𝜒 , 𝜃 , 𝜑 ) ↔ if- ( 𝜒 , 𝜃 , 𝜓 ) ) )