Theorem sylanr2 696

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 629	. 2 ⊢ ((𝜒 ∧ 𝜑) → (𝜒 ∧ 𝜃))
3		sylanr2.2	. 2 ⊢ ((𝜓 ∧ (𝜒 ∧ 𝜃)) → 𝜏)
4	2, 3	sylan2 605	1 ⊢ ((𝜓 ∧ (𝜒 ∧ 𝜑)) → 𝜏)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-an 402

This theorem is used by:  adantrrl  737  adantrrr  738  unfi  9190  isfin7-2  10478  mulsub  11763  fzsubel  13698  expsub  14257  ramlb  17203  0ram  17204  cmprmidlmcl  21637  ressmplvsca  22345  tgcl  23293  fgss2  24199  nmoid  25067  madebdaylemlrcut  28291  numclwwlkqhash  30982  chirredlem4  33001  pibt2  38434  lindsadd  38630  poimirlem28  38660  pridlc3  39101  stoweidlem34  47138