Theorem wl-luk-pm2.27 35533

Description: This theorem, called "Assertion", can be thought of as closed form of modus ponens ax-mp 5. Theorem *2.27 of [WhiteheadRussell] p. 104. Copy of pm2.27 42 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

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

Proof of Theorem wl-luk-pm2.27

Step	Hyp	Ref	Expression
1		wl-luk-ax1 35532	. . 3 ⊢ (𝜑 → (¬ 𝜓 → 𝜑))
2		ax-luk1 35517	. . 3 ⊢ ((¬ 𝜓 → 𝜑) → ((𝜑 → 𝜓) → (¬ 𝜓 → 𝜓)))
3	1, 2	wl-luk-syl 35522	. 2 ⊢ (𝜑 → ((𝜑 → 𝜓) → (¬ 𝜓 → 𝜓)))
4		ax-luk2 35518	. 2 ⊢ ((¬ 𝜓 → 𝜓) → 𝜓)
5	3, 4	wl-luk-imtrdi 35530	1 ⊢ (𝜑 → ((𝜑 → 𝜓) → 𝜓))

Syntax hints:  ¬ wn 3   → wi 4

This theorem was proved from axioms:  ax-mp 5  ax-luk1 35517  ax-luk2 35518  ax-luk3 35519

This theorem is referenced by:  wl-luk-com12  35534