Theorem pm2.18d 127

Description: Deduction form of the Clavius law pm2.18 128. (Contributed by FL, 12-Jul-2009.) (Proof shortened by Andrew Salmon, 7-May-2011.) Revised to shorten pm2.18 128. (Revised by Wolf Lammen, 17-Nov-2023.)

Hypothesis

Ref	Expression
pm2.18d.1	⊢ (𝜑 → (¬ 𝜓 → 𝜓))

Assertion

Ref	Expression
pm2.18d	⊢ (𝜑 → 𝜓)

Proof of Theorem pm2.18d

Step	Hyp	Ref	Expression
1		id 22	. 2 ⊢ (𝜑 → 𝜑)
2		pm2.18d.1	. . 3 ⊢ (𝜑 → (¬ 𝜓 → 𝜓))
3		pm2.21 123	. . 3 ⊢ (¬ 𝜓 → (𝜓 → ¬ 𝜑))
4	2, 3	sylcom 30	. 2 ⊢ (𝜑 → (¬ 𝜓 → ¬ 𝜑))
5	1, 4	mt4d 117	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:  pm2.18  128  pm2.61d  179  pm2.18da  799  oplem1  1056  axc11n  2426  weniso  7351  infssuni  9343  ordtypelem10  9522  oismo  9535  rankval3b  9821  grur1  10815  sqeqd  15113  hausflimi  23484  minveclem4  24949  ovolunnul  25017  vitali  25130  itg2mono  25271  frgrncvvdeqlem8  29559  minvecolem4  30133  contrd  36965  fppr2odd  46399