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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 12984. (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 12947 | . 2 ⊢ BOUNDED [𝑦 / 𝑥]𝜑 |
3 | sbsbc 2886 | . 2 ⊢ ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑) | |
4 | 2, 3 | bd0 12949 | 1 ⊢ BOUNDED [𝑦 / 𝑥]𝜑 |
Colors of variables: wff set class |
Syntax hints: [wsb 1720 [wsbc 2882 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-5 1408 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-4 1472 ax-17 1491 ax-ial 1499 ax-ext 2099 ax-bd0 12938 ax-bdsb 12947 |
This theorem depends on definitions: df-bi 116 df-clab 2104 df-cleq 2110 df-clel 2113 df-sbc 2883 |
This theorem is referenced by: bdccsb 12985 |
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