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Theorem pm2.65d 187
 Description: Deduction rule for proof by contradiction. (Contributed by NM, 26-Jun-1994.) (Proof shortened by Wolf Lammen, 26-May-2013.)
Hypotheses
Ref Expression
pm2.65d.1 (𝜑 → (𝜓𝜒))
pm2.65d.2 (𝜑 → (𝜓 → ¬ 𝜒))
Assertion
Ref Expression
pm2.65d (𝜑 → ¬ 𝜓)

Proof of Theorem pm2.65d
StepHypRef Expression
1 pm2.65d.2 . . 3 (𝜑 → (𝜓 → ¬ 𝜒))
2 pm2.65d.1 . . 3 (𝜑 → (𝜓𝜒))
31, 2nsyld 154 . 2 (𝜑 → (𝜓 → ¬ 𝜓))
43pm2.01d 181 1 (𝜑 → ¬ 𝜓)
 Colors of variables: wff setvar class 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:  mtod  189  pm2.65da  599  unxpdomlem2  8206  cardlim  8836  winainflem  9553  winalim2  9556  discr  13041  sqrmo  14036  vdwnnlem3  15748  nmlno0lem  27776  nmlnop0iALT  28982  iooelexlt  33340
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