Theorem sylan2i 687

Description: A syllogism inference. (Contributed by NM, 1-Aug-1994.)

Hypotheses

Ref	Expression
sylan2i.1	⊢ (𝜑 → 𝜃)
sylan2i.2	⊢ (𝜓 → ((𝜒 ∧ 𝜃) → 𝜏))

Assertion

Ref	Expression
sylan2i	⊢ (𝜓 → ((𝜒 ∧ 𝜑) → 𝜏))

Proof of Theorem sylan2i

Step	Hyp	Ref	Expression
1		sylan2i.1	. . 3 ⊢ (𝜑 → 𝜃)
2	1	a1i 11	. 2 ⊢ (𝜓 → (𝜑 → 𝜃))
3		sylan2i.2	. 2 ⊢ (𝜓 → ((𝜒 ∧ 𝜃) → 𝜏))
4	2, 3	sylan2d 499	1 ⊢ (𝜓 → ((𝜒 ∧ 𝜑) → 𝜏))

Syntax hints:   → wi 4   ∧ wa 384

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 197  df-an 386

This theorem is referenced by:  syl2ani  688  odi  7659  pssnn  8178  ltexprlem7  9864  ltaprlem  9866  sup2  10979  filufint  21724  pjnormssi  29027  poimirlem27  33436  poimirlem31  33440  pellex  37399