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Theorem syl7bi 244
 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 205 . 2 (𝜑𝜓)
3 syl7bi.2 . 2 (𝜒 → (𝜃 → (𝜓𝜏)))
42, 3syl7 72 1 (𝜒 → (𝜃 → (𝜑𝜏)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195 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 196 This theorem is referenced by:  rspct  3275  zfpair  4831  gruen  9513  axpre-sup  9869  nn0lt2  11317  fzofzim  12382  ndvdssub  14971  alexsubALT  21665  clwlkisclwwlklem2a  26313  erclwwlktr  26343  erclwwlkntr  26355  dfon2lem8  30939  bj-nfimt  32025  prtlem15  33178  prtlem18  33180  clwlkclwwlklem2a  41207  erclwwlkstr  41243  erclwwlksntr  41255
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