Theorem wl-syl5 32423

Description: A syllogism rule of inference. The first premise is used to replace the second antecedent of the second premise. Copy of syl5 33 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
wl-syl5.1	⊢ (𝜑 → 𝜓)
wl-syl5.2	⊢ (𝜒 → (𝜓 → 𝜃))

Assertion

Ref	Expression
wl-syl5	⊢ (𝜒 → (𝜑 → 𝜃))

Proof of Theorem wl-syl5

Step	Hyp	Ref	Expression
1		wl-syl5.2	. 2 ⊢ (𝜒 → (𝜓 → 𝜃))
2		wl-syl5.1	. . 3 ⊢ (𝜑 → 𝜓)
3	2	wl-imim1i 32421	. 2 ⊢ ((𝜓 → 𝜃) → (𝜑 → 𝜃))
4	1, 3	wl-syl 32422	1 ⊢ (𝜒 → (𝜑 → 𝜃))

Syntax hints:   → wi 4

This theorem was proved from axioms:  ax-mp 5  ax-luk1 32417

This theorem is referenced by:  wl-con4i  32425  wl-mpi  32428  wl-ax3  32431  wl-com12  32434  wl-con1i  32436  wl-ja  32437  wl-pm2.04  32443