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Theorem syl7bi 255
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 216 . 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 206
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 207
This theorem is referenced by:  3jao  1426  rspct  3607  zfpair  5420  gruen  10853  axpre-sup  11210  nn0lt2  12683  fzofzim  13750  ndvdssub  16447  cyccom  19222  alexsubALT  24060  clwlkclwwlklem2a  30018  erclwwlktr  30042  erclwwlkntr  30091  fmlasuc  35392  dfon2lem8  35792  prtlem15  38877  prtlem18  38879  2reuimp0  47131
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