Theorem imimorb 847

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		bi2.04 350	. 2 ⊢ (((ψ → χ) → (φ → χ)) ↔ (φ → ((ψ → χ) → χ)))
2		dfor2 400	. . 3 ⊢ ((ψ ∨ χ) ↔ ((ψ → χ) → χ))
3	2	imbi2i 303	. 2 ⊢ ((φ → (ψ ∨ χ)) ↔ (φ → ((ψ → χ) → χ)))
4	1, 3	bitr4i 243	1 ⊢ (((ψ → χ) → (φ → χ)) ↔ (φ → (ψ ∨ χ)))

Syntax hints:   → wi 4   ↔ wb 176   ∨ wo 357

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 177  df-or 359

This theorem is referenced by: (None)