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Theorem biantr 971
Description: A transitive law of equivalence. Compare Theorem *4.22 of [WhiteheadRussell] p. 117. (Contributed by NM, 18-Aug-1993.)
Assertion
Ref Expression
biantr (((𝜑𝜓) ∧ (𝜒𝜓)) → (𝜑𝜒))

Proof of Theorem biantr
StepHypRef Expression
1 id 22 . . 3 ((𝜒𝜓) → (𝜒𝜓))
21bibi2d 332 . 2 ((𝜒𝜓) → ((𝜑𝜒) ↔ (𝜑𝜓)))
32biimparc 504 1 (((𝜑𝜓) ∧ (𝜒𝜓)) → (𝜑𝜒))
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:  bm1.1  2605  bitr3VD  38904  sbcoreleleqVD  38915  trsbcVD  38933  sbcssgVD  38939
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