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	1	imbi2d	⊢ ( ( ( 𝜑 ↔ 𝜓 ) ∧ ( 𝜒 ↔ 𝜃 ) ) → ( ( 𝜏 → 𝜑 ) ↔ ( 𝜏 → 𝜓 ) ) )
3		simpr	⊢ ( ( ( 𝜑 ↔ 𝜓 ) ∧ ( 𝜒 ↔ 𝜃 ) ) → ( 𝜒 ↔ 𝜃 ) )
4	3	imbi2d	⊢ ( ( ( 𝜑 ↔ 𝜓 ) ∧ ( 𝜒 ↔ 𝜃 ) ) → ( ( ¬ 𝜏 → 𝜒 ) ↔ ( ¬ 𝜏 → 𝜃 ) ) )
5	2 4	anbi12d	⊢ ( ( ( 𝜑 ↔ 𝜓 ) ∧ ( 𝜒 ↔ 𝜃 ) ) → ( ( ( 𝜏 → 𝜑 ) ∧ ( ¬ 𝜏 → 𝜒 ) ) ↔ ( ( 𝜏 → 𝜓 ) ∧ ( ¬ 𝜏 → 𝜃 ) ) ) )
6		dfifp2	⊢ ( if- ( 𝜏 , 𝜑 , 𝜒 ) ↔ ( ( 𝜏 → 𝜑 ) ∧ ( ¬ 𝜏 → 𝜒 ) ) )
7		dfifp2	⊢ ( if- ( 𝜏 , 𝜓 , 𝜃 ) ↔ ( ( 𝜏 → 𝜓 ) ∧ ( ¬ 𝜏 → 𝜃 ) ) )
8	5 6 7	3bitr4g	⊢ ( ( ( 𝜑 ↔ 𝜓 ) ∧ ( 𝜒 ↔ 𝜃 ) ) → ( if- ( 𝜏 , 𝜑 , 𝜒 ) ↔ if- ( 𝜏 , 𝜓 , 𝜃 ) ) )