Theorem dfor2 199

Description: Logical 'or' expressed in terms of implication only. Theorem *5.25 of [WhiteheadRussell] p. 124.

Assertion

Ref	Expression
dfor2	⊢ ((φ ∨ ψ) ↔ ((φ → ψ) → ψ))

Proof of Theorem dfor2

Step	Hyp	Ref	Expression
1		df-or 197	. 2 ⊢ ((φ ∨ ψ) ↔ (¬ φ → ψ))
2		pm2.61 109	. . . 4 ⊢ ((φ → ψ) → ((¬ φ → ψ) → ψ))
3	2	com12 13	. . 3 ⊢ ((¬ φ → ψ) → ((φ → ψ) → ψ))
4		pm2.21 71	. . . 4 ⊢ (¬ φ → (φ → ψ))
5	4	syl4 19	. . 3 ⊢ (((φ → ψ) → ψ) → (¬ φ → ψ))
6	3, 5	impbi 139	. 2 ⊢ ((¬ φ → ψ) ↔ ((φ → ψ) → ψ))
7	1, 6	bitr 151	1 ⊢ ((φ ∨ ψ) ↔ ((φ → ψ) → ψ))

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

This theorem is referenced by:  pm2.62 210

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

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