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Theorem syl6rbb 186
Description: A syllogism inference from two biconditionals. (Contributed by NM, 5-Aug-1993.)
Hypotheses
Ref Expression
syl6rbb.1 (𝜑 → (𝜓𝜒))
syl6rbb.2 (𝜒𝜃)
Assertion
Ref Expression
syl6rbb (𝜑 → (𝜃𝜓))

Proof of Theorem syl6rbb
StepHypRef Expression
1 syl6rbb.1 . . 3 (𝜑 → (𝜓𝜒))
2 syl6rbb.2 . . 3 (𝜒𝜃)
31, 2syl6bb 185 . 2 (𝜑 → (𝜓𝜃))
43bicomd 129 1 (𝜑 → (𝜃𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101
This theorem depends on definitions:  df-bi 110
This theorem is referenced by:  syl6rbbr  188  bibif  614  pm5.61  708  oranabs  728  pm5.7dc  861  nbbndc  1285  resopab2  4655  xpcom  4864  ac6sfi  6352  elznn0  8258  rexuz3  9562
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