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Theorem syl3an3 1170
 Description: A syllogism inference. (Contributed by NM, 22-Aug-1995.)
Hypotheses
Ref Expression
syl3an3.1 (𝜑𝜃)
syl3an3.2 ((𝜓𝜒𝜃) → 𝜏)
Assertion
Ref Expression
syl3an3 ((𝜓𝜒𝜑) → 𝜏)

Proof of Theorem syl3an3
StepHypRef Expression
1 syl3an3.1 . . 3 (𝜑𝜃)
2 syl3an3.2 . . . 4 ((𝜓𝜒𝜃) → 𝜏)
323exp 1103 . . 3 (𝜓 → (𝜒 → (𝜃𝜏)))
41, 3syl7 63 . 2 (𝜓 → (𝜒 → (𝜑𝜏)))
543imp 1098 1 ((𝜓𝜒𝜑) → 𝜏)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ w3a 885 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101 This theorem depends on definitions:  df-bi 110  df-3an 887 This theorem is referenced by:  syl3an3b  1173  syl3an3br  1176  vtoclgft  2604  ovmpt2x  5629  ovmpt2ga  5630  nnanq0  6556  apreim  7594  divassap  7669  ltmul2  7822  elfzo  9006  subcn2  9832  mulcn2  9833
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