Theorem orcana 509

Description: Disjunction in consequent versus conjunction in antecedent. Similar to Theorem *5.6 of [WhiteheadRussell] p. 125.

Assertion

Ref	Expression
orcana	⊢ ((φ → (ψ ∨ χ)) ↔ ((φ ∧ ¬ ψ) → χ))

Proof of Theorem orcana

Step	Hyp	Ref	Expression
1		df-or 197	. . 3 ⊢ ((ψ ∨ χ) ↔ (¬ ψ → χ))
2	1	imbi2i 160	. 2 ⊢ ((φ → (ψ ∨ χ)) ↔ (φ → (¬ ψ → χ)))
3		impexp 276	. 2 ⊢ (((φ ∧ ¬ ψ) → χ) ↔ (φ → (¬ ψ → χ)))
4	2, 3	bitr4 154	1 ⊢ ((φ → (ψ ∨ χ)) ↔ ((φ ∧ ¬ ψ) → χ))

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   ∨ wo 195   ∧ wa 196

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198

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