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Theorem syl2ani 687
Description: A syllogism inference. (Contributed by NM, 3-Aug-1999.)
Hypotheses
Ref Expression
syl2ani.1 (𝜑𝜒)
syl2ani.2 (𝜂𝜃)
syl2ani.3 (𝜓 → ((𝜒𝜃) → 𝜏))
Assertion
Ref Expression
syl2ani (𝜓 → ((𝜑𝜂) → 𝜏))

Proof of Theorem syl2ani
StepHypRef Expression
1 syl2ani.1 . 2 (𝜑𝜒)
2 syl2ani.2 . . 3 (𝜂𝜃)
3 syl2ani.3 . . 3 (𝜓 → ((𝜒𝜃) → 𝜏))
42, 3sylan2i 686 . 2 (𝜓 → ((𝜒𝜂) → 𝜏))
51, 4sylani 685 1 (𝜓 → ((𝜑𝜂) → 𝜏))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384
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 197  df-an 386
This theorem is referenced by:  2mo  2550  frxp  7232  mapen  8068  fin1a2lem9  9174  coprmproddvdslem  15300  psss  17135  mgmidmo  17180  aannenlem1  23987  funtransport  31777  cgrxfr  31801  btwnxfr  31802  bj-cbv3tb  32350
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