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Theorem syl7bi 243
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 204 . 2 (𝜑𝜓)
3 syl7bi.2 . 2 (𝜒 → (𝜃 → (𝜓𝜏)))
42, 3syl7 71 1 (𝜒 → (𝜃 → (𝜑𝜏)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194
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 195
This theorem is referenced by:  rspct  3271  zfpair  4823  gruen  9487  axpre-sup  9843  nn0lt2  11270  fzofzim  12334  ndvdssub  14914  alexsubALT  21604  clwlkisclwwlklem2a  26076  erclwwlktr  26106  erclwwlkntr  26118  dfon2lem8  30742  prtlem15  32978  prtlem18  32980  clwlkclwwlklem2a  41206  erclwwlkstr  41242  erclwwlksntr  41254
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