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Theorem syland 604
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 417 . . 3 (𝜑 → (𝜒 → (𝜃𝜏)))
41, 3syld 47 . 2 (𝜑 → (𝜓 → (𝜃𝜏)))
54impd 412 1 (𝜑 → ((𝜓𝜃) → 𝜏))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397
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 398
This theorem is referenced by:  sylani  605  sylan2d  606  syl2and  609  onfununi  8341  lt2add  11699  nn0seqcvgd  16507  1stcelcls  22965  llyidm  22992  filuni  23389  ballotlemimin  33535  btwnintr  35022  ifscgr  35047  btwnconn1lem12  35101  poimir  36569  cvrntr  38344  goldbachthlem2  46262
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