Theorem pm2.65 654

Description: Theorem *2.65 of [WhiteheadRussell] p. 107. Proof by contradiction. Proofs, such as this one, which assume a proposition, here 𝜑, derive a contradiction, and therefore conclude ¬ 𝜑, are valid intuitionistically (and can be called "proof of negation", for example by Section 1.2 of [Bauer] p. 482). By contrast, proofs which assume ¬ 𝜑, derive a contradiction, and conclude 𝜑, such as condandc 876, are only valid for decidable propositions. (Contributed by NM, 5-Aug-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		pm2.27 40	. . . 4 ⊢ (𝜑 → ((𝜑 → ¬ 𝜓) → ¬ 𝜓))
2	1	con2d 619	. . 3 ⊢ (𝜑 → (𝜓 → ¬ (𝜑 → ¬ 𝜓)))
3	2	a2i 11	. 2 ⊢ ((𝜑 → 𝜓) → (𝜑 → ¬ (𝜑 → ¬ 𝜓)))
4	3	con2d 619	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-in1 609  ax-in2 610

This theorem is referenced by:  pm4.82  945