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Theorem sylanr1 680
Description: A syllogism inference. (Contributed by NM, 9-Apr-2005.)
Hypotheses
Ref Expression
sylanr1.1 (𝜑𝜒)
sylanr1.2 ((𝜓 ∧ (𝜒𝜃)) → 𝜏)
Assertion
Ref Expression
sylanr1 ((𝜓 ∧ (𝜑𝜃)) → 𝜏)

Proof of Theorem sylanr1
StepHypRef Expression
1 sylanr1.1 . . 3 (𝜑𝜒)
21anim1i 613 . 2 ((𝜑𝜃) → (𝜒𝜃))
3 sylanr1.2 . 2 ((𝜓 ∧ (𝜒𝜃)) → 𝜏)
42, 3sylan2 591 1 ((𝜓 ∧ (𝜑𝜃)) → 𝜏)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394
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 206  df-an 395
This theorem is referenced by:  adantrll  720  adantrlr  721  sbthlem9  9118  unfi  9199  pczpre  16841  cpmadugsumlemF  22863  blsscls2  24498  rpvmasumlem  27510  leopmuli  32060  chirredlem1  32317  chirredlem3  32319  pibt2  37134  mhpind  42281  dvconstbi  44042  bccbc  44053  reccot  48537  rectan  48538  aacllem  48582
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