Theorem pm4.82 741

Description: Theorem *4.82 of [WhiteheadRussell] p. 122.

Assertion

Ref	Expression
pm4.82	⊢ (((φ → ψ) ⋀ (φ → ¬ ψ)) ↔ ¬ φ)

Proof of Theorem pm4.82

Step	Hyp	Ref	Expression
1		pm2.65 134	. . 3 ⊢ ((φ → ψ) → ((φ → ¬ ψ) → ¬ φ))
2	1	imp 350	. 2 ⊢ (((φ → ψ) ⋀ (φ → ¬ ψ)) → ¬ φ)
3		pm2.21 76	. . 3 ⊢ (¬ φ → (φ → ψ))
4		pm2.21 76	. . 3 ⊢ (¬ φ → (φ → ¬ ψ))
5	3, 4	jca 288	. 2 ⊢ (¬ φ → ((φ → ψ) ⋀ (φ → ¬ ψ)))
6	2, 5	impbi 157	1 ⊢ (((φ → ψ) ⋀ (φ → ¬ ψ)) ↔ ¬ φ)

Syntax hints:  ¬ wn 2   → wi 3   ↔ wb 146   ⋀ wa 223

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7

This theorem depends on definitions:  df-bi 147  df-an 225

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