Theorem wl-luk-syl 38180

Description: An inference version of the transitive laws for implication luk-1 1688. Copy of syl 18 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.)

Hypotheses

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

Assertion

Ref	Expression
wl-luk-syl	⊢ (𝜑 → 𝜒)

Proof of Theorem wl-luk-syl

Step	Hyp	Ref	Expression
1		wl-luk-syl.2	. 2 ⊢ (𝜓 → 𝜒)
2		wl-luk-syl.1	. . 3 ⊢ (𝜑 → 𝜓)
3	2	wl-luk-imim1i 38179	. 2 ⊢ ((𝜓 → 𝜒) → (𝜑 → 𝜒))
4	1, 3	ax-mp 5	1 ⊢ (𝜑 → 𝜒)

This proof depends on syntax axioms:   → wi 4

This proof depends on axioms:  ax-mp 5  ax-luk1 38175

This theorem is used by:  wl-luk-imtrid  38181  wl-luk-pm2.18d  38182  wl-luk-imtrdi  38188  wl-luk-ax1  38190  wl-luk-pm2.27  38191  wl-luk-a1d  38197  wl-luk-id  38199