Theorem frege45

Description: Deduce pm2.6 from con1 . Proposition 45 of Frege1879 p. 47. (Contributed by RP, 24-Dec-2019) (Proof modification is discouraged.)

	Ref	Expression
Assertion	frege45	⊢ ( ( ( ¬ 𝜑 → 𝜓 ) → ( ¬ 𝜓 → 𝜑 ) ) → ( ( ¬ 𝜑 → 𝜓 ) → ( ( 𝜑 → 𝜓 ) → 𝜓 ) ) )

Proof

Step	Hyp	Ref	Expression
1		frege44	⊢ ( ( ¬ 𝜓 → 𝜑 ) → ( ( 𝜑 → 𝜓 ) → 𝜓 ) )
2		frege5	⊢ ( ( ( ¬ 𝜓 → 𝜑 ) → ( ( 𝜑 → 𝜓 ) → 𝜓 ) ) → ( ( ( ¬ 𝜑 → 𝜓 ) → ( ¬ 𝜓 → 𝜑 ) ) → ( ( ¬ 𝜑 → 𝜓 ) → ( ( 𝜑 → 𝜓 ) → 𝜓 ) ) ) )
3	1 2	ax-mp	⊢ ( ( ( ¬ 𝜑 → 𝜓 ) → ( ¬ 𝜓 → 𝜑 ) ) → ( ( ¬ 𝜑 → 𝜓 ) → ( ( 𝜑 → 𝜓 ) → 𝜓 ) ) )