Theorem pm2.65 192

Description: Theorem *2.65 of [WhiteheadRussell] p. 107. Proof by contradiction. (Contributed by NM, 21-Jun-1993.) (Proof shortened by Wolf Lammen, 8-Mar-2013.)

Assertion

Ref	Expression
pm2.65	⊢ ((𝜑 → 𝜓) → ((𝜑 → ¬ 𝜓) → ¬ 𝜑))

Proof of Theorem pm2.65

Step	Hyp	Ref	Expression
1		idd 24	. 2 ⊢ ((𝜑 → 𝜓) → (¬ 𝜑 → ¬ 𝜑))
2		con3 153	. 2 ⊢ ((𝜑 → 𝜓) → (¬ 𝜓 → ¬ 𝜑))
3	1, 2	jad 187	1 ⊢ ((𝜑 → 𝜓) → ((𝜑 → ¬ 𝜓) → ¬ 𝜑))

Syntax hints:  ¬ wn 3   → wi 4

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem is referenced by:  pm4.82  1020