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Mirrors > Home > ILE Home > Th. List > pm2.65  Structured version GIF version 
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, 5Aug1993.) (Proof shortened by Wolf Lammen, 8Mar2013.) 
Ref  Expression 

pm2.65  ⊢ ((φ → ψ) → ((φ → ¬ ψ) → ¬ φ)) 
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)) 
Colors of variables: wff set class 
Syntax hints: ¬ wn 3 → wi 4 
This theorem was proved from axioms: ax1 5 ax2 6 axmp 7 axin1 544 axin2 545 
This theorem is referenced by: pm4.82 856 
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