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Theorem sylanblc 587
Description: Syllogism inference combined with a biconditional. (Contributed by BJ, 25-Apr-2019.)
Hypotheses
Ref Expression
sylanblc.1 (𝜑𝜓)
sylanblc.2 𝜒
sylanblc.3 ((𝜓𝜒) ↔ 𝜃)
Assertion
Ref Expression
sylanblc (𝜑𝜃)

Proof of Theorem sylanblc
StepHypRef Expression
1 sylanblc.1 . 2 (𝜑𝜓)
2 sylanblc.2 . 2 𝜒
3 sylanblc.3 . . 3 ((𝜓𝜒) ↔ 𝜃)
43biimpi 215 . 2 ((𝜓𝜒) → 𝜃)
51, 2, 4sylancl 584 1 (𝜑𝜃)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394
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 206  df-an 395
This theorem is referenced by:  uniintsn  4988  xmulpnf1  13299  odd2np1  16336  eltg3i  22950  restntr  23172  cmpcld  23392  rnelfm  23943  ovolctb2  25507  noextendseq  27692  iscgra  28731  isinag  28760  isleag  28769  iseqlg  28789  omlsilem  31330  mblfinlem3  37371
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