Theorem wl-luk-imtrid 38268

Description: A syllogism rule of inference. The first premise is used to replace the second antecedent of the second premise. Copy of syl5 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-imtrid.1	⊢ (𝜑 → 𝜓)
wl-luk-imtrid.2	⊢ (𝜒 → (𝜓 → 𝜃))

Assertion

Ref	Expression
wl-luk-imtrid	⊢ (𝜒 → (𝜑 → 𝜃))

Proof of Theorem wl-luk-imtrid

Step	Hyp	Ref	Expression
1		wl-luk-imtrid.2	. 2 ⊢ (𝜒 → (𝜓 → 𝜃))
2		wl-luk-imtrid.1	. . 3 ⊢ (𝜑 → 𝜓)
3	2	wl-luk-imim1i 38266	. 2 ⊢ ((𝜓 → 𝜃) → (𝜑 → 𝜃))
4	1, 3	wl-luk-syl 38267	1 ⊢ (𝜒 → (𝜑 → 𝜃))

This proof depends on syntax axioms:   → wi 4

This proof depends on axioms:  ax-mp 5  ax-luk1 38262

This theorem is used by:  wl-luk-con4i  38270  wl-luk-mpi  38273  wl-luk-ax3  38276  wl-luk-com12  38279  wl-luk-con1i  38281  wl-luk-ja  38282  wl-luk-pm2.04  38288