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Theorem bdsbc 12890
Description: A formula resulting from proper substitution of a setvar for a setvar in a bounded formula is bounded. See also bdsbcALT 12891. (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 12854 . 2 BOUNDED [𝑦 / 𝑥]𝜑
3 sbsbc 2884 . 2 ([𝑦 / 𝑥]𝜑[𝑦 / 𝑥]𝜑)
42, 3bd0 12856 1 BOUNDED [𝑦 / 𝑥]𝜑
Colors of variables: wff set class
Syntax hints:  [wsb 1718  [wsbc 2880  BOUNDED wbd 12844
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 1406  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-4 1470  ax-17 1489  ax-ial 1497  ax-ext 2097  ax-bd0 12845  ax-bdsb 12854
This theorem depends on definitions:  df-bi 116  df-clab 2102  df-cleq 2108  df-clel 2111  df-sbc 2881
This theorem is referenced by:  bdccsb  12892
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