Theorem imimorb 917

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

Assertion

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

Proof of Theorem imimorb

Step	Hyp	Ref	Expression
1		bi2.04 375	. 2 ⊢ (((𝜓 → 𝜒) → (𝜑 → 𝜒)) ↔ (𝜑 → ((𝜓 → 𝜒) → 𝜒)))
2		dfor2 426	. . 3 ⊢ ((𝜓 ∨ 𝜒) ↔ ((𝜓 → 𝜒) → 𝜒))
3	2	imbi2i 325	. 2 ⊢ ((𝜑 → (𝜓 ∨ 𝜒)) ↔ (𝜑 → ((𝜓 → 𝜒) → 𝜒)))
4	1, 3	bitr4i 266	1 ⊢ (((𝜓 → 𝜒) → (𝜑 → 𝜒)) ↔ (𝜑 → (𝜓 ∨ 𝜒)))

Syntax hints:   → wi 4   ↔ wb 195   ∨ wo 382

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 196  df-or 384

This theorem is referenced by: (None)