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Metamath Proof Explorer

Theorem pm2.18d

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

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

Proof

Step	Hyp	Ref	Expression
1		pm2.18d.1	⊢ ( 𝜑 → ( ¬ 𝜓 → 𝜓 ) )
2		id	⊢ ( 𝜑 → 𝜑 )
3		pm2.21	⊢ ( ¬ 𝜓 → ( 𝜓 → ¬ 𝜑 ) )
4	1 3	sylcom	⊢ ( 𝜑 → ( ¬ 𝜓 → ¬ 𝜑 ) )
5	2 4	mt4d	⊢ ( 𝜑 → 𝜓 )