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Theorem syland 603
Description: A syllogism deduction. (Contributed by NM, 15-Dec-2004.)
Hypotheses
Ref Expression
syland.1 (𝜑 → (𝜓𝜒))
syland.2 (𝜑 → ((𝜒𝜃) → 𝜏))
Assertion
Ref Expression
syland (𝜑 → ((𝜓𝜃) → 𝜏))

Proof of Theorem syland
StepHypRef Expression
1 syland.1 . . 3 (𝜑 → (𝜓𝜒))
2 syland.2 . . . 4 (𝜑 → ((𝜒𝜃) → 𝜏))
32expd 415 . . 3 (𝜑 → (𝜒 → (𝜃𝜏)))
41, 3syld 47 . 2 (𝜑 → (𝜓 → (𝜃𝜏)))
54impd 410 1 (𝜑 → ((𝜓𝜃) → 𝜏))
Colors of variables: wff setvar class
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:  sylani  604  sylan2d  605  syl2and  608  onfununi  8381  fodomfir  9368  lt2add  11748  nn0seqcvgd  16607  1stcelcls  23469  llyidm  23496  filuni  23893  ballotlemimin  34508  btwnintr  36020  ifscgr  36045  btwnconn1lem12  36099  poimir  37660  cvrntr  39427  goldbachthlem2  47533
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