Theorem sylanr2 685

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 593	. 2 ⊢ ((𝜒 ∧ 𝜑) → (𝜒 ∧ 𝜃))
3		sylanr2.2	. 2 ⊢ ((𝜓 ∧ (𝜒 ∧ 𝜃)) → 𝜏)
4	2, 3	sylan2 491	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:  adantrrl  760  adantrrr  761  1stconst  7265  2ndconst  7266  isfin7-2  9218  mulsub  10473  fzsubel  12377  expsub  12908  ramlb  15723  0ram  15724  ressmplvsca  19459  tgcl  20773  fgss2  21678  nmoid  22546  chirredlem4  29252  poimirlem28  33437  pridlc3  33872  stoweidlem34  40251