Theorem wl-pm2.27 33257

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-pm2.27	⊢ (𝜑 → ((𝜑 → 𝜓) → 𝜓))

Proof of Theorem wl-pm2.27

Step	Hyp	Ref	Expression
1		wl-ax1 33256	. . 3 ⊢ (𝜑 → (¬ 𝜓 → 𝜑))
2		ax-luk1 33241	. . 3 ⊢ ((¬ 𝜓 → 𝜑) → ((𝜑 → 𝜓) → (¬ 𝜓 → 𝜓)))
3	1, 2	wl-syl 33246	. 2 ⊢ (𝜑 → ((𝜑 → 𝜓) → (¬ 𝜓 → 𝜓)))
4		ax-luk2 33242	. 2 ⊢ ((¬ 𝜓 → 𝜓) → 𝜓)
5	3, 4	wl-syl6 33254	1 ⊢ (𝜑 → ((𝜑 → 𝜓) → 𝜓))

Syntax hints:  ¬ wn 3   → wi 4

This theorem was proved from axioms:  ax-mp 5  ax-luk1 33241  ax-luk2 33242  ax-luk3 33243

This theorem is referenced by:  wl-com12  33258