Theorem sylani 604

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 603	1 ⊢ (𝜓 → ((𝜑 ∧ 𝜃) → 𝜏))

Syntax hints:   → wi 4   ∧ wa 396

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 206  df-an 397

This theorem is referenced by:  syl2ani  607  inf3lem2  9387  zorn2lem5  10256  uzwo  12651  supxrun  13050  lcmdvds  16313  cramer0  21839  csmdsymi  30696  matunitlindflem2  35774  pmapjoin  37866