Theorem bd3an 14667

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

Step	Hyp	Ref	Expression
1		bd3or.1	. . . 4 ⊢ BOUNDED 𝜑
2		bd3or.2	. . . 4 ⊢ BOUNDED 𝜓
3	1, 2	ax-bdan 14652	. . 3 ⊢ BOUNDED (𝜑 ∧ 𝜓)
4		bd3or.3	. . 3 ⊢ BOUNDED 𝜒
5	3, 4	ax-bdan 14652	. 2 ⊢ BOUNDED ((𝜑 ∧ 𝜓) ∧ 𝜒)
6		df-3an 980	. 2 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ((𝜑 ∧ 𝜓) ∧ 𝜒))
7	5, 6	bd0r 14662	1 ⊢ BOUNDED (𝜑 ∧ 𝜓 ∧ 𝜒)

Syntax hints:   ∧ wa 104   ∧ w3a 978  BOUNDED wbd 14649

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-bd0 14650  ax-bdan 14652

This theorem depends on definitions:  df-bi 117  df-3an 980

This theorem is referenced by: (None)