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Theorem bd3an 9950
 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 (𝜑𝜓𝜒)

Proof of Theorem bd3an
StepHypRef Expression
1 bd3or.1 . . . 4 BOUNDED 𝜑
2 bd3or.2 . . . 4 BOUNDED 𝜓
31, 2ax-bdan 9935 . . 3 BOUNDED (𝜑𝜓)
4 bd3or.3 . . 3 BOUNDED 𝜒
53, 4ax-bdan 9935 . 2 BOUNDED ((𝜑𝜓) ∧ 𝜒)
6 df-3an 887 . 2 ((𝜑𝜓𝜒) ↔ ((𝜑𝜓) ∧ 𝜒))
75, 6bd0r 9945 1 BOUNDED (𝜑𝜓𝜒)
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   ∧ w3a 885  BOUNDED wbd 9932 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 9933  ax-bdan 9935 This theorem depends on definitions:  df-bi 110  df-3an 887 This theorem is referenced by: (None)
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