Theorem imimorb 965

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		bi2.04 392	. 2 ⊢ (((𝜓 → 𝜒) → (𝜑 → 𝜒)) ↔ (𝜑 → ((𝜓 → 𝜒) → 𝜒)))
2		dfor2 915	. . 3 ⊢ ((𝜓 ∨ 𝜒) ↔ ((𝜓 → 𝜒) → 𝜒))
3	2	imbi2i 339	. 2 ⊢ ((𝜑 → (𝜓 ∨ 𝜒)) ↔ (𝜑 → ((𝜓 → 𝜒) → 𝜒)))
4	1, 3	bitr4i 281	1 ⊢ (((𝜓 → 𝜒) → (𝜑 → 𝜒)) ↔ (𝜑 → (𝜓 ∨ 𝜒)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∨ wo 861

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-or 862

This theorem is used by: (None)