Theorem pm2.65 584

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 774, 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 35	. . . 4 ⊢ (φ → ((φ → ¬ ψ) → ¬ ψ))
2	1	con2d 554	. . 3 ⊢ (φ → (ψ → ¬ (φ → ¬ ψ)))
3	2	a2i 11	. 2 ⊢ ((φ → ψ) → (φ → ¬ (φ → ¬ ψ)))
4	3	con2d 554	1 ⊢ ((φ → ψ) → ((φ → ¬ ψ) → ¬ φ))

Syntax hints:  ¬ wn 3   → wi 4

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-in1 544  ax-in2 545

This theorem is referenced by:  pm4.82  856