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Mirrors > Home > ILE Home > Th. List > Mathboxes > bdsbc | GIF version |
Description: A formula resulting from proper substitution of a setvar for a setvar in a bounded formula is bounded. See also bdsbcALT 13228. (Contributed by BJ, 16-Oct-2019.) |
Ref | Expression |
---|---|
bdcsbc.1 | ⊢ BOUNDED 𝜑 |
Ref | Expression |
---|---|
bdsbc | ⊢ BOUNDED [𝑦 / 𝑥]𝜑 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bdcsbc.1 | . . 3 ⊢ BOUNDED 𝜑 | |
2 | 1 | ax-bdsb 13191 | . 2 ⊢ BOUNDED [𝑦 / 𝑥]𝜑 |
3 | sbsbc 2917 | . 2 ⊢ ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑) | |
4 | 2, 3 | bd0 13193 | 1 ⊢ BOUNDED [𝑦 / 𝑥]𝜑 |
Colors of variables: wff set class |
Syntax hints: [wsb 1736 [wsbc 2913 BOUNDED wbd 13181 |
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-5 1424 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-4 1488 ax-17 1507 ax-ial 1515 ax-ext 2122 ax-bd0 13182 ax-bdsb 13191 |
This theorem depends on definitions: df-bi 116 df-clab 2127 df-cleq 2133 df-clel 2136 df-sbc 2914 |
This theorem is referenced by: bdccsb 13229 |
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