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)

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:  pm2.18  129  pm2.61d  181  pm2.18da  812  oplem1  1072  axc11n  2461  weniso  7365  infssuni  9313  ordtypelem10  9499  oismo  9512  rankval3b  9808  grur1  10823  sqeqd  15243  hausflimi  24174  minveclem4  25628  ovolunnul  25696  vitali  25809  itg2mono  25949  frgrncvvdeqlem8  30694  minvecolem4  31269  contrd  38787  fppr2odd  48537