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Theorem bdsbc 9978
 Description: A formula resulting from proper substitution of a setvar for a setvar in a bounded formula is bounded. See also bdsbcALT 9979. (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 9942 . 2 BOUNDED [𝑦 / 𝑥]𝜑
3 sbsbc 2768 . 2 ([𝑦 / 𝑥]𝜑[𝑦 / 𝑥]𝜑)
42, 3bd0 9944 1 BOUNDED [𝑦 / 𝑥]𝜑
 Colors of variables: wff set class Syntax hints:  [wsb 1645  [wsbc 2764  BOUNDED wbd 9932 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-17 1419  ax-ial 1427  ax-ext 2022  ax-bd0 9933  ax-bdsb 9942 This theorem depends on definitions:  df-bi 110  df-clab 2027  df-cleq 2033  df-clel 2036  df-sbc 2765 This theorem is referenced by:  bdccsb  9980
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