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Theorem syl7bi 258
 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 219 . 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 209 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 210 This theorem is referenced by:  3jao  1422  rspct  3595  zfpair  5310  gruen  10228  axpre-sup  10585  nn0lt2  12040  fzofzim  13086  ndvdssub  15756  cyccom  18344  alexsubALT  22654  clwlkclwwlklem2a  27781  erclwwlktr  27805  erclwwlkntr  27854  fmlasuc  32660  dfon2lem8  33062  prtlem15  36083  prtlem18  36085  2reuimp0  43536
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