Mathbox for BJ < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >   Mathboxes  >  bd3an Structured version   GIF version

Theorem bd3an 9265
 Description: A conjunction of three bounded formulas is bounded. (Contributed by BJ, 3-Oct-2019.)
Hypotheses
Ref Expression
bd3or.1 BOUNDED φ
bd3or.2 BOUNDED ψ
bd3or.3 BOUNDED χ
Assertion
Ref Expression
bd3an BOUNDED (φ ψ χ)

Proof of Theorem bd3an
StepHypRef Expression
1 bd3or.1 . . . 4 BOUNDED φ
2 bd3or.2 . . . 4 BOUNDED ψ
31, 2ax-bdan 9250 . . 3 BOUNDED (φ ψ)
4 bd3or.3 . . 3 BOUNDED χ
53, 4ax-bdan 9250 . 2 BOUNDED ((φ ψ) χ)
6 df-3an 886 . 2 ((φ ψ χ) ↔ ((φ ψ) χ))
75, 6bd0r 9260 1 BOUNDED (φ ψ χ)
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   ∧ w3a 884  BOUNDED wbd 9247 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-bd0 9248  ax-bdan 9250 This theorem depends on definitions:  df-bi 110  df-3an 886 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator