Theorem wl-luk-pm2.27

Description: This theorem, called "Assertion", can be thought of as closed form of modus ponens ax-mp . Theorem *2.27 of WhiteheadRussell p. 104. Copy of pm2.27 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018) (New usage is discouraged.) (Proof modification is discouraged.)

	Ref	Expression
Assertion	wl-luk-pm2.27	⊢ ( 𝜑 → ( ( 𝜑 → 𝜓 ) → 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		wl-luk-ax1	⊢ ( 𝜑 → ( ¬ 𝜓 → 𝜑 ) )
2		ax-luk1	⊢ ( ( ¬ 𝜓 → 𝜑 ) → ( ( 𝜑 → 𝜓 ) → ( ¬ 𝜓 → 𝜓 ) ) )
3	1 2	wl-luk-syl	⊢ ( 𝜑 → ( ( 𝜑 → 𝜓 ) → ( ¬ 𝜓 → 𝜓 ) ) )
4		ax-luk2	⊢ ( ( ¬ 𝜓 → 𝜓 ) → 𝜓 )
5	3 4	wl-luk-imtrdi	⊢ ( 𝜑 → ( ( 𝜑 → 𝜓 ) → 𝜓 ) )