Theorem ifpbi13

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

	Ref	Expression
Assertion	ifpbi13	⊢ ( ( ( 𝜑 ↔ 𝜓 ) ∧ ( 𝜒 ↔ 𝜃 ) ) → ( if- ( 𝜑 , 𝜏 , 𝜒 ) ↔ if- ( 𝜓 , 𝜏 , 𝜃 ) ) )

Proof

Step	Hyp	Ref	Expression
1		simpl	⊢ ( ( ( 𝜑 ↔ 𝜓 ) ∧ ( 𝜒 ↔ 𝜃 ) ) → ( 𝜑 ↔ 𝜓 ) )
2	1	imbi1d	⊢ ( ( ( 𝜑 ↔ 𝜓 ) ∧ ( 𝜒 ↔ 𝜃 ) ) → ( ( 𝜑 → 𝜏 ) ↔ ( 𝜓 → 𝜏 ) ) )
3		notbi	⊢ ( ( 𝜑 ↔ 𝜓 ) ↔ ( ¬ 𝜑 ↔ ¬ 𝜓 ) )
4		imbi12	⊢ ( ( ¬ 𝜑 ↔ ¬ 𝜓 ) → ( ( 𝜒 ↔ 𝜃 ) → ( ( ¬ 𝜑 → 𝜒 ) ↔ ( ¬ 𝜓 → 𝜃 ) ) ) )
5	3 4	sylbi	⊢ ( ( 𝜑 ↔ 𝜓 ) → ( ( 𝜒 ↔ 𝜃 ) → ( ( ¬ 𝜑 → 𝜒 ) ↔ ( ¬ 𝜓 → 𝜃 ) ) ) )
6	5	imp	⊢ ( ( ( 𝜑 ↔ 𝜓 ) ∧ ( 𝜒 ↔ 𝜃 ) ) → ( ( ¬ 𝜑 → 𝜒 ) ↔ ( ¬ 𝜓 → 𝜃 ) ) )
7	2 6	anbi12d	⊢ ( ( ( 𝜑 ↔ 𝜓 ) ∧ ( 𝜒 ↔ 𝜃 ) ) → ( ( ( 𝜑 → 𝜏 ) ∧ ( ¬ 𝜑 → 𝜒 ) ) ↔ ( ( 𝜓 → 𝜏 ) ∧ ( ¬ 𝜓 → 𝜃 ) ) ) )
8		dfifp2	⊢ ( if- ( 𝜑 , 𝜏 , 𝜒 ) ↔ ( ( 𝜑 → 𝜏 ) ∧ ( ¬ 𝜑 → 𝜒 ) ) )
9		dfifp2	⊢ ( if- ( 𝜓 , 𝜏 , 𝜃 ) ↔ ( ( 𝜓 → 𝜏 ) ∧ ( ¬ 𝜓 → 𝜃 ) ) )
10	7 8 9	3bitr4g	⊢ ( ( ( 𝜑 ↔ 𝜓 ) ∧ ( 𝜒 ↔ 𝜃 ) ) → ( if- ( 𝜑 , 𝜏 , 𝜒 ) ↔ if- ( 𝜓 , 𝜏 , 𝜃 ) ) )