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Theorem bdsbc 11106
Description: A formula resulting from proper substitution of a setvar for a setvar in a bounded formula is bounded. See also bdsbcALT 11107. (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 11070 . 2 BOUNDED [𝑦 / 𝑥]𝜑
3 sbsbc 2832 . 2 ([𝑦 / 𝑥]𝜑[𝑦 / 𝑥]𝜑)
42, 3bd0 11072 1 BOUNDED [𝑦 / 𝑥]𝜑
Colors of variables: wff set class
Syntax hints:  [wsb 1689  [wsbc 2828  BOUNDED wbd 11060
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1379  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-4 1443  ax-17 1462  ax-ial 1470  ax-ext 2067  ax-bd0 11061  ax-bdsb 11070
This theorem depends on definitions:  df-bi 115  df-clab 2072  df-cleq 2078  df-clel 2081  df-sbc 2829
This theorem is referenced by:  bdccsb  11108
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