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Mirrors > Home > ILE Home > Th. List > Mathboxes > bds | Unicode version |
Description: Boundedness of a formula resulting from implicit substitution in a bounded formula. Note that the proof does not use ax-bdsb 10613; therefore, using implicit instead of explicit substitution when boundedness is important, one might avoid using ax-bdsb 10613. (Contributed by BJ, 19-Nov-2019.) |
Ref | Expression |
---|---|
bds.bd | BOUNDED |
bds.1 |
Ref | Expression |
---|---|
bds | BOUNDED |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bds.bd | . . . 4 BOUNDED | |
2 | 1 | bdcab 10640 | . . 3 BOUNDED |
3 | bds.1 | . . . 4 | |
4 | 3 | cbvabv 2202 | . . 3 |
5 | 2, 4 | bdceqi 10634 | . 2 BOUNDED |
6 | 5 | bdph 10641 | 1 BOUNDED |
Colors of variables: wff set class |
Syntax hints: wi 4 wb 103 cab 2067 BOUNDED wbd 10603 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-bd0 10604 ax-bdsb 10613 |
This theorem depends on definitions: df-bi 115 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-bdc 10632 |
This theorem is referenced by: (None) |
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