Theorem pm2.18d 128

Description: Deduction form of the Clavius law pm2.18 129. (Contributed by FL, 12-Jul-2009.) (Proof shortened by Andrew Salmon, 7-May-2011.) Shorten pm2.18 129. (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 23	. 2 ⊢ (𝜑 → 𝜑)
2		pm2.18d.1	. . 3 ⊢ (𝜑 → (¬ 𝜓 → 𝜓))
3		pm2.21 124	. . 3 ⊢ (¬ 𝜓 → (𝜓 → ¬ 𝜑))
4	2, 3	sylcom 31	. 2 ⊢ (𝜑 → (¬ 𝜓 → ¬ 𝜑))
5	1, 4	mt4d 118	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  129  pm2.61d  181  pm2.18da  811  oplem1  1072  axc11n  2458  weniso  7354  infssuni  9304  ordtypelem10  9490  oismo  9503  rankval3b  9799  grur1  10806  sqeqd  15219  hausflimi  24118  minveclem4  25572  ovolunnul  25640  vitali  25753  itg2mono  25893  frgrncvvdeqlem8  30635  minvecolem4  31210  contrd  38724  fppr2odd  48473