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 617	. 2 ⊢ ((𝜒 ∧ 𝜑) → (𝜒 ∧ 𝜃))
3		sylanr2.2	. 2 ⊢ ((𝜓 ∧ (𝜒 ∧ 𝜃)) → 𝜏)
4	2, 3	sylan2 593	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:  adantrrl  724  adantrrr  725  unfi  9135  isfin7-2  10349  mulsub  11621  fzsubel  13521  expsub  14075  ramlb  16990  0ram  16991  ressmplvsca  21938  tgcl  22856  fgss2  23761  nmoid  24630  madebdaylemlrcut  27810  numclwwlkqhash  30304  chirredlem4  32322  pibt2  37405  lindsadd  37607  poimirlem28  37642  pridlc3  38067  stoweidlem34  46032