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Mirrors > Home > ILE Home > Th. List > Mathboxes > bdceq | GIF version |
Description: Equality property for the predicate BOUNDED. (Contributed by BJ, 3-Oct-2019.) |
Ref | Expression |
---|---|
bdceq.1 | ⊢ 𝐴 = 𝐵 |
Ref | Expression |
---|---|
bdceq | ⊢ (BOUNDED 𝐴 ↔ BOUNDED 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bdceq.1 | . . . . 5 ⊢ 𝐴 = 𝐵 | |
2 | 1 | eleq2i 2244 | . . . 4 ⊢ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) |
3 | 2 | bdeq 14545 | . . 3 ⊢ (BOUNDED 𝑥 ∈ 𝐴 ↔ BOUNDED 𝑥 ∈ 𝐵) |
4 | 3 | albii 1470 | . 2 ⊢ (∀𝑥BOUNDED 𝑥 ∈ 𝐴 ↔ ∀𝑥BOUNDED 𝑥 ∈ 𝐵) |
5 | df-bdc 14563 | . 2 ⊢ (BOUNDED 𝐴 ↔ ∀𝑥BOUNDED 𝑥 ∈ 𝐴) | |
6 | df-bdc 14563 | . 2 ⊢ (BOUNDED 𝐵 ↔ ∀𝑥BOUNDED 𝑥 ∈ 𝐵) | |
7 | 4, 5, 6 | 3bitr4i 212 | 1 ⊢ (BOUNDED 𝐴 ↔ BOUNDED 𝐵) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 105 ∀wal 1351 = wceq 1353 ∈ wcel 2148 BOUNDED wbd 14534 BOUNDED wbdc 14562 |
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-5 1447 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-4 1510 ax-17 1526 ax-ial 1534 ax-ext 2159 ax-bd0 14535 |
This theorem depends on definitions: df-bi 117 df-cleq 2170 df-clel 2173 df-bdc 14563 |
This theorem is referenced by: bdceqi 14565 |
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