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Theorem bdsbc 12983
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.)
Hypothesis
Ref Expression
bdcsbc.1 BOUNDED 𝜑
Assertion
Ref Expression
bdsbc BOUNDED [𝑦 / 𝑥]𝜑

Proof of Theorem bdsbc
StepHypRef Expression
1 bdcsbc.1 . . 3 BOUNDED 𝜑
21ax-bdsb 12947 . 2 BOUNDED [𝑦 / 𝑥]𝜑
3 sbsbc 2886 . 2 ([𝑦 / 𝑥]𝜑[𝑦 / 𝑥]𝜑)
42, 3bd0 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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