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Theorem sylanblrc 696
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 693 1 (𝜑𝜃)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384
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 386
This theorem is referenced by:  fnwelem  7277  tfrlem10  7468  gruina  9625  dfac14  21402  1trld  26982  1stmbfm  30296  2ndmbfm  30297  bj-projval  32959  rfcnpre1  38998
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