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Theorem syl7bi 256
Description: A mixed syllogism inference from a doubly nested implication and a biconditional. (Contributed by NM, 14-May-1993.)
Hypotheses
Ref Expression
syl7bi.1 (𝜑𝜓)
syl7bi.2 (𝜒 → (𝜃 → (𝜓𝜏)))
Assertion
Ref Expression
syl7bi (𝜒 → (𝜃 → (𝜑𝜏)))

Proof of Theorem syl7bi
StepHypRef Expression
1 syl7bi.1 . . 3 (𝜑𝜓)
21biimpi 217 . 2 (𝜑𝜓)
3 syl7bi.2 . 2 (𝜒 → (𝜃 → (𝜓𝜏)))
42, 3syl7 74 1 (𝜒 → (𝜃 → (𝜑𝜏)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207
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 208
This theorem is referenced by:  3jao  1417  rspct  3606  zfpair  5312  gruen  10222  axpre-sup  10579  nn0lt2  12033  fzofzim  13072  ndvdssub  15748  cyccom  18284  alexsubALT  22587  clwlkclwwlklem2a  27703  erclwwlktr  27727  erclwwlkntr  27777  fmlasuc  32530  dfon2lem8  32932  prtlem15  35891  prtlem18  35893  2reuimp0  43190
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