Theorem frege46

Description: If ps holds when ph occurs as well as when ph does not occur, then ps holds. If ps or ph occurs and if the occurrences of ph has ps as a necessary consequence, then ps takes place. Identical to pm2.6 . Proposition 46 of Frege1879 p. 48. (Contributed by RP, 24-Dec-2019) (Proof modification is discouraged.)

	Ref	Expression
Assertion	frege46	⊢ ( ( ¬ 𝜑 → 𝜓 ) → ( ( 𝜑 → 𝜓 ) → 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		frege33	⊢ ( ( ¬ 𝜑 → 𝜓 ) → ( ¬ 𝜓 → 𝜑 ) )
2		frege45	⊢ ( ( ( ¬ 𝜑 → 𝜓 ) → ( ¬ 𝜓 → 𝜑 ) ) → ( ( ¬ 𝜑 → 𝜓 ) → ( ( 𝜑 → 𝜓 ) → 𝜓 ) ) )
3	1 2	ax-mp	⊢ ( ( ¬ 𝜑 → 𝜓 ) → ( ( 𝜑 → 𝜓 ) → 𝜓 ) )