Theorem imimorb

Description: Simplify an implication between implications. (Contributed by Paul Chapman, 17-Nov-2012) (Proof shortened by Wolf Lammen, 3-Apr-2013)

	Ref	Expression
Assertion	imimorb	⊢ ( ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ↔ ( 𝜑 → ( 𝜓 ∨ 𝜒 ) ) )

Proof

Step	Hyp	Ref	Expression
1		bi2.04	⊢ ( ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ↔ ( 𝜑 → ( ( 𝜓 → 𝜒 ) → 𝜒 ) ) )
2		dfor2	⊢ ( ( 𝜓 ∨ 𝜒 ) ↔ ( ( 𝜓 → 𝜒 ) → 𝜒 ) )
3	2	imbi2i	⊢ ( ( 𝜑 → ( 𝜓 ∨ 𝜒 ) ) ↔ ( 𝜑 → ( ( 𝜓 → 𝜒 ) → 𝜒 ) ) )
4	1 3	bitr4i	⊢ ( ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ↔ ( 𝜑 → ( 𝜓 ∨ 𝜒 ) ) )