Theorem pm2.65d 199

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 157	. 2 ⊢ (𝜑 → (𝜓 → ¬ 𝜓))
4	3	pm2.01d 192	1 ⊢ (𝜑 → ¬ 𝜓)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem is used by:  mtod  201  pm2.65da  829  unxpdomlem2  9226  cardlim  10025  winainflem  10750  winalim2  10753  discr  14352  sqrmo  15386  vdwnnlem3  17137  psdmul  22449  nmlno0lem  31329  nmlnop0iALT  32531  iooelexlt  38205