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Theorem syl9r 71
Description: A nested syllogism inference with different antecedents. (Contributed by NM, 5-Aug-1993.)
Hypotheses
Ref Expression
syl9r.1 (𝜑 → (𝜓𝜒))
syl9r.2 (𝜃 → (𝜒𝜏))
Assertion
Ref Expression
syl9r (𝜃 → (𝜑 → (𝜓𝜏)))

Proof of Theorem syl9r
StepHypRef Expression
1 syl9r.1 . . 3 (𝜑 → (𝜓𝜒))
2 syl9r.2 . . 3 (𝜃 → (𝜒𝜏))
31, 2syl9 70 . 2 (𝜑 → (𝜃 → (𝜓𝜏)))
43com12 30 1 (𝜃 → (𝜑 → (𝜓𝜏)))
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:  sylan9r  396  pm2.85dc  822  looinvdc  832  pclem6  1281  nfimd  1493  19.23t  1583  fununi  4995  dfimafn  5250  funimass3  5311  nnsub  8028
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