Theorem sylani 616

Description: A syllogism inference. (Contributed by NM, 2-May-1996.)

Hypotheses

Ref	Expression
sylani.1	⊢ (𝜑 → 𝜒)
sylani.2	⊢ (𝜓 → ((𝜒 ∧ 𝜃) → 𝜏))

Assertion

Ref	Expression
sylani	⊢ (𝜓 → ((𝜑 ∧ 𝜃) → 𝜏))

Proof of Theorem sylani

Step	Hyp	Ref	Expression
1		sylani.1	. . 3 ⊢ (𝜑 → 𝜒)
2	1	a1i 11	. 2 ⊢ (𝜓 → (𝜑 → 𝜒))
3		sylani.2	. 2 ⊢ (𝜓 → ((𝜒 ∧ 𝜃) → 𝜏))
4	2, 3	syland 615	1 ⊢ (𝜓 → ((𝜑 ∧ 𝜃) → 𝜏))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-an 402

This theorem is used by:  syl2ani  619  inf3lem2  9667  r1filimi  9940  zorn2lem5  10617  uzwo  13078  supxrun  13486  lcmdvds  16827  matunitlindflem2  23050  cramer0  23063  csmdsymi  32993  acwer1prclem  35892  pmapjoin  41175