Theorem sylan2i 606

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 605	1 ⊢ (𝜓 → ((𝜒 ∧ 𝜑) → 𝜏))

Syntax hints:   → wi 4   ∧ wa 395

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 207  df-an 396

This theorem is referenced by:  syl2ani  607  odi  8617  pssnn  9208  ltexprlem7  11082  ltaprlem  11084  sup2  12224  filufint  23928  pjnormssi  32187  poimirlem27  37654  poimirlem31  37658  sn-sup2  42501  pellex  42846