Theorem ifpbi1

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

	Ref	Expression
Assertion	ifpbi1	⊢ ( ( 𝜑 ↔ 𝜓 ) → ( if- ( 𝜑 , 𝜒 , 𝜃 ) ↔ if- ( 𝜓 , 𝜒 , 𝜃 ) ) )

Proof

Step	Hyp	Ref	Expression
1		imbi1	⊢ ( ( 𝜑 ↔ 𝜓 ) → ( ( 𝜑 → 𝜒 ) ↔ ( 𝜓 → 𝜒 ) ) )
2		notbi	⊢ ( ( 𝜑 ↔ 𝜓 ) ↔ ( ¬ 𝜑 ↔ ¬ 𝜓 ) )
3	2	biimpi	⊢ ( ( 𝜑 ↔ 𝜓 ) → ( ¬ 𝜑 ↔ ¬ 𝜓 ) )
4	3	imbi1d	⊢ ( ( 𝜑 ↔ 𝜓 ) → ( ( ¬ 𝜑 → 𝜃 ) ↔ ( ¬ 𝜓 → 𝜃 ) ) )
5	1 4	anbi12d	⊢ ( ( 𝜑 ↔ 𝜓 ) → ( ( ( 𝜑 → 𝜒 ) ∧ ( ¬ 𝜑 → 𝜃 ) ) ↔ ( ( 𝜓 → 𝜒 ) ∧ ( ¬ 𝜓 → 𝜃 ) ) ) )
6		dfifp2	⊢ ( if- ( 𝜑 , 𝜒 , 𝜃 ) ↔ ( ( 𝜑 → 𝜒 ) ∧ ( ¬ 𝜑 → 𝜃 ) ) )
7		dfifp2	⊢ ( if- ( 𝜓 , 𝜒 , 𝜃 ) ↔ ( ( 𝜓 → 𝜒 ) ∧ ( ¬ 𝜓 → 𝜃 ) ) )
8	5 6 7	3bitr4g	⊢ ( ( 𝜑 ↔ 𝜓 ) → ( if- ( 𝜑 , 𝜒 , 𝜃 ) ↔ if- ( 𝜓 , 𝜒 , 𝜃 ) ) )