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Theorem syl8 70
Description: A syllogism rule of inference. The second premise is used to replace the consequent of the first premise. (Contributed by NM, 1-Aug-1994.) (Proof shortened by Wolf Lammen, 3-Aug-2012.)
Hypotheses
Ref Expression
syl8.1 (𝜑 → (𝜓 → (𝜒𝜃)))
syl8.2 (𝜃𝜏)
Assertion
Ref Expression
syl8 (𝜑 → (𝜓 → (𝜒𝜏)))

Proof of Theorem syl8
StepHypRef Expression
1 syl8.1 . 2 (𝜑 → (𝜓 → (𝜒𝜃)))
2 syl8.2 . . 3 (𝜃𝜏)
32a1i 9 . 2 (𝜑 → (𝜃𝜏))
41, 3syl6d 69 1 (𝜑 → (𝜓 → (𝜒𝜏)))
Colors of variables: wff set class
Syntax hints:  wi 4
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7
This theorem is referenced by:  com45  88  syl8ib  164  imp5a  350  con4biddc  790  3exp  1140  suctr  4222  ssorduni  4277  nneneq  6525  qreccl  9059  bj-inf2vnlem2  11311
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