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Theorem sylanblrc 570
Description: Syllogism inference combined with a biconditional. (Contributed by BJ, 25-Apr-2019.)
Hypotheses
Ref Expression
sylanblrc.1 (𝜑𝜓)
sylanblrc.2 𝜒
sylanblrc.3 (𝜃 ↔ (𝜓𝜒))
Assertion
Ref Expression
sylanblrc (𝜑𝜃)

Proof of Theorem sylanblrc
StepHypRef Expression
1 sylanblrc.1 . 2 (𝜑𝜓)
2 sylanblrc.2 . 2 𝜒
3 sylanblrc.3 . . 3 (𝜃 ↔ (𝜓𝜒))
43biimpri 218 . 2 ((𝜓𝜒) → 𝜃)
51, 2, 4sylancl 566 1 (𝜑𝜃)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382
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 197  df-an 383
This theorem is referenced by:  fnwelem  7442  tfrlem10  7635  gruina  9841  dfac14  21641  1trld  27319  1stmbfm  30656  2ndmbfm  30657  bj-projval  33309  rfcnpre1  39694
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