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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 12947; therefore, using implicit instead of explicit substitution when boundedness is important, one might avoid using ax-bdsb 12947. (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 12974 | . . 3 BOUNDED |
3 | bds.1 | . . . 4 | |
4 | 3 | cbvabv 2241 | . . 3 |
5 | 2, 4 | bdceqi 12968 | . 2 BOUNDED |
6 | 5 | bdph 12975 | 1 BOUNDED |
Colors of variables: wff set class |
Syntax hints: wi 4 wb 104 cab 2103 BOUNDED wbd 12937 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-bd0 12938 ax-bdsb 12947 |
This theorem depends on definitions: df-bi 116 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-bdc 12966 |
This theorem is referenced by: (None) |
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