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Theorem bitr2id 193
Description: A syllogism inference from two biconditionals. (Contributed by NM, 5-Aug-1993.)
Hypotheses
Ref Expression
bitr2id.1 (𝜑𝜓)
bitr2id.2 (𝜒 → (𝜓𝜃))
Assertion
Ref Expression
bitr2id (𝜒 → (𝜃𝜑))

Proof of Theorem bitr2id
StepHypRef Expression
1 bitr2id.1 . . 3 (𝜑𝜓)
2 bitr2id.2 . . 3 (𝜒 → (𝜓𝜃))
31, 2bitrid 192 . 2 (𝜒 → (𝜑𝜃))
43bicomd 141 1 (𝜒 → (𝜃𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108
This theorem depends on definitions:  df-bi 117
This theorem is referenced by:  bitr3di  195  pm5.17dc  904  dn1dc  960  csbabg  3119  uniiunlem  3245  inimasn  5047  cnvpom  5172  fnresdisj  5327  f1oiso  5827  reldm  6187  mptelixpg  6734  1idprl  7589  1idpru  7590  nndiv  8960  fzn  10042  fz1sbc  10096  grpid  12912  metrest  14009
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