Mathbox for BJ |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > Mathboxes > bd3an | GIF version |
Description: A conjunction of three bounded formulas is bounded. (Contributed by BJ, 3-Oct-2019.) |
Ref | Expression |
---|---|
bd3or.1 | ⊢ BOUNDED 𝜑 |
bd3or.2 | ⊢ BOUNDED 𝜓 |
bd3or.3 | ⊢ BOUNDED 𝜒 |
Ref | Expression |
---|---|
bd3an | ⊢ BOUNDED (𝜑 ∧ 𝜓 ∧ 𝜒) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bd3or.1 | . . . 4 ⊢ BOUNDED 𝜑 | |
2 | bd3or.2 | . . . 4 ⊢ BOUNDED 𝜓 | |
3 | 1, 2 | ax-bdan 9935 | . . 3 ⊢ BOUNDED (𝜑 ∧ 𝜓) |
4 | bd3or.3 | . . 3 ⊢ BOUNDED 𝜒 | |
5 | 3, 4 | ax-bdan 9935 | . 2 ⊢ BOUNDED ((𝜑 ∧ 𝜓) ∧ 𝜒) |
6 | df-3an 887 | . 2 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ((𝜑 ∧ 𝜓) ∧ 𝜒)) | |
7 | 5, 6 | bd0r 9945 | 1 ⊢ BOUNDED (𝜑 ∧ 𝜓 ∧ 𝜒) |
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) |
Copyright terms: Public domain | W3C validator |