Theorem wl-luk-imtrdi 35509

Description: A syllogism rule of inference. The second premise is used to replace the consequent of the first premise. Copy of syl6 35 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
wl-luk-imtrdi.1	⊢ (𝜑 → (𝜓 → 𝜒))
wl-luk-imtrdi.2	⊢ (𝜒 → 𝜃)

Assertion

Ref	Expression
wl-luk-imtrdi	⊢ (𝜑 → (𝜓 → 𝜃))

Proof of Theorem wl-luk-imtrdi

Step	Hyp	Ref	Expression
1		wl-luk-imtrdi.1	. 2 ⊢ (𝜑 → (𝜓 → 𝜒))
2		wl-luk-imtrdi.2	. . 3 ⊢ (𝜒 → 𝜃)
3	2	wl-luk-imim2i 35508	. 2 ⊢ ((𝜓 → 𝜒) → (𝜓 → 𝜃))
4	1, 3	wl-luk-syl 35501	1 ⊢ (𝜑 → (𝜓 → 𝜃))

Syntax hints:   → wi 4

This theorem was proved from axioms:  ax-mp 5  ax-luk1 35496  ax-luk2 35497  ax-luk3 35498

This theorem is referenced by:  wl-luk-ax3  35510  wl-luk-pm2.27  35512