Theorem pm2.65d 195

Description: Deduction 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 → (𝜓 → 𝜒))
pm2.65d.2	⊢ (𝜑 → (𝜓 → ¬ 𝜒))

Assertion

Ref	Expression
pm2.65d	⊢ (𝜑 → ¬ 𝜓)

Proof of Theorem pm2.65d

Step	Hyp	Ref	Expression
1		pm2.65d.2	. . 3 ⊢ (𝜑 → (𝜓 → ¬ 𝜒))
2		pm2.65d.1	. . 3 ⊢ (𝜑 → (𝜓 → 𝜒))
3	1, 2	nsyld 156	. 2 ⊢ (𝜑 → (𝜓 → ¬ 𝜓))
4	3	pm2.01d 189	1 ⊢ (𝜑 → ¬ 𝜓)

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  197  pm2.65da  813  unxpdomlem2  8957  cardlim  9661  winainflem  10380  winalim2  10383  discr  13883  sqrmo  14891  vdwnnlem3  16626  nmlno0lem  29056  nmlnop0iALT  30258  iooelexlt  35460