Theorem sylanr2 665

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 605	. 2 ⊢ ((𝜒 ∧ 𝜑) → (𝜒 ∧ 𝜃))
3		sylanr2.2	. 2 ⊢ ((𝜓 ∧ (𝜒 ∧ 𝜃)) → 𝜏)
4	2, 3	sylan2 582	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 198  df-an 385

This theorem is referenced by:  adantrrl  706  adantrrr  707  1stconst  7493  2ndconst  7494  isfin7-2  9497  mulsub  10752  fzsubel  12594  expsub  13125  ramlb  15934  0ram  15935  ressmplvsca  19662  tgcl  20981  fgss2  21885  nmoid  22753  numclwwlkqhash  27549  chirredlem4  29574  poimirlem28  33744  pridlc3  34177  stoweidlem34  40724