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Theorem sylcom 28
Description: Syllogism inference with commutation of antecedents. (Contributed by NM, 29-Aug-2004.) (Proof shortened by O'Cat, 2-Feb-2006.) (Proof shortened by Stefan Allan, 23-Feb-2006.)
Hypotheses
Ref Expression
sylcom.1 (𝜑 → (𝜓𝜒))
sylcom.2 (𝜓 → (𝜒𝜃))
Assertion
Ref Expression
sylcom (𝜑 → (𝜓𝜃))

Proof of Theorem sylcom
StepHypRef Expression
1 sylcom.1 . 2 (𝜑 → (𝜓𝜒))
2 sylcom.2 . . 3 (𝜓 → (𝜒𝜃))
32a2i 11 . 2 ((𝜓𝜒) → (𝜓𝜃))
41, 3syl 14 1 (𝜑 → (𝜓𝜃))
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:  syl5com  29  syl6  33  syli  37  mpbidi  149  con4biddc  788  jaddc  795  con1biddc  804  necon4addc  2316  necon4bddc  2317  necon4ddc  2318  necon1addc  2322  necon1bddc  2323  dmcosseq  4631  iss  4684  funopg  4964  snon0  6445
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