Theorem imbi12

Description: Closed form of imbi12i . Was automatically derived from its "Virtual Deduction" version and the Metamath program "MM-PA> MINIMIZE__WITH *" command. (Contributed by Alan Sare, 18-Mar-2012)

	Ref	Expression
Assertion	imbi12	⊢ ( ( 𝜑 ↔ 𝜓 ) → ( ( 𝜒 ↔ 𝜃 ) → ( ( 𝜑 → 𝜒 ) ↔ ( 𝜓 → 𝜃 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		simplim	⊢ ( ¬ ( ( 𝜑 ↔ 𝜓 ) → ¬ ( 𝜒 ↔ 𝜃 ) ) → ( 𝜑 ↔ 𝜓 ) )
2		simprim	⊢ ( ¬ ( ( 𝜑 ↔ 𝜓 ) → ¬ ( 𝜒 ↔ 𝜃 ) ) → ( 𝜒 ↔ 𝜃 ) )
3	1 2	imbi12d	⊢ ( ¬ ( ( 𝜑 ↔ 𝜓 ) → ¬ ( 𝜒 ↔ 𝜃 ) ) → ( ( 𝜑 → 𝜒 ) ↔ ( 𝜓 → 𝜃 ) ) )
4	3	expi	⊢ ( ( 𝜑 ↔ 𝜓 ) → ( ( 𝜒 ↔ 𝜃 ) → ( ( 𝜑 → 𝜒 ) ↔ ( 𝜓 → 𝜃 ) ) ) )