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Theorem syl2and 295
Description: A syllogism deduction. (Contributed by NM, 15-Dec-2004.)
Hypotheses
Ref Expression
syl2and.1 (𝜑 → (𝜓𝜒))
syl2and.2 (𝜑 → (𝜃𝜏))
syl2and.3 (𝜑 → ((𝜒𝜏) → 𝜂))
Assertion
Ref Expression
syl2and (𝜑 → ((𝜓𝜃) → 𝜂))

Proof of Theorem syl2and
StepHypRef Expression
1 syl2and.1 . 2 (𝜑 → (𝜓𝜒))
2 syl2and.2 . . 3 (𝜑 → (𝜃𝜏))
3 syl2and.3 . . 3 (𝜑 → ((𝜒𝜏) → 𝜂))
42, 3sylan2d 294 . 2 (𝜑 → ((𝜒𝜃) → 𝜂))
51, 4syland 293 1 (𝜑 → ((𝜓𝜃) → 𝜂))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108
This theorem depends on definitions:  df-bi 117
This theorem is referenced by:  anim12d  335  recexprlem1ssl  7632  recexprlem1ssu  7633  xle2add  9879  fzen  10043  bezoutlembi  12006  rpmulgcd2  12095  pcqmul  12303  2sqlem8a  14472
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