Theorem sylanr2 683

Description: A syllogism inference. (Contributed by NM, 9-Apr-2005.)

Hypotheses

Ref	Expression
sylanr2.1	⊢ (𝜑 → 𝜃)
sylanr2.2	⊢ ((𝜓 ∧ (𝜒 ∧ 𝜃)) → 𝜏)

Assertion

Ref	Expression
sylanr2	⊢ ((𝜓 ∧ (𝜒 ∧ 𝜑)) → 𝜏)

Proof of Theorem sylanr2

Step	Hyp	Ref	Expression
1		sylanr2.1	. . 3 ⊢ (𝜑 → 𝜃)
2	1	anim2i 591	. 2 ⊢ ((𝜒 ∧ 𝜑) → (𝜒 ∧ 𝜃))
3		sylanr2.2	. 2 ⊢ ((𝜓 ∧ (𝜒 ∧ 𝜃)) → 𝜏)
4	2, 3	sylan2 490	1 ⊢ ((𝜓 ∧ (𝜒 ∧ 𝜑)) → 𝜏)

Syntax hints:   → wi 4   ∧ wa 383

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 196  df-an 385

This theorem is referenced by:  adantrrl  756  adantrrr  757  1stconst  7152  2ndconst  7153  isfin7-2  9101  mulsub  10352  fzsubel  12248  expsub  12770  ramlb  15561  0ram  15562  ressmplvsca  19280  tgcl  20584  fgss2  21488  nmoid  22356  chirredlem4  28636  poimirlem28  32607  pridlc3  33042  stoweidlem34  38927