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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 15008. (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 14971 | . 2 ⊢ BOUNDED [𝑦 / 𝑥]𝜑 |
3 | sbsbc 2981 | . 2 ⊢ ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑) | |
4 | 2, 3 | bd0 14973 | 1 ⊢ BOUNDED [𝑦 / 𝑥]𝜑 |
Colors of variables: wff set class |
Syntax hints: [wsb 1773 [wsbc 2977 BOUNDED wbd 14961 |
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 1458 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-4 1521 ax-17 1537 ax-ial 1545 ax-ext 2171 ax-bd0 14962 ax-bdsb 14971 |
This theorem depends on definitions: df-bi 117 df-clab 2176 df-cleq 2182 df-clel 2185 df-sbc 2978 |
This theorem is referenced by: bdccsb 15009 |
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