Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >   Mathboxes  >  bdsbc GIF version

Theorem bdsbc 13893
Description: A formula resulting from proper substitution of a setvar for a setvar in a bounded formula is bounded. See also bdsbcALT 13894. (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 13857 . 2 BOUNDED [𝑦 / 𝑥]𝜑
3 sbsbc 2959 . 2 ([𝑦 / 𝑥]𝜑[𝑦 / 𝑥]𝜑)
42, 3bd0 13859 1 BOUNDED [𝑦 / 𝑥]𝜑
Colors of variables: wff set class
Syntax hints:  [wsb 1755  [wsbc 2955  BOUNDED wbd 13847
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 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503  ax-17 1519  ax-ial 1527  ax-ext 2152  ax-bd0 13848  ax-bdsb 13857
This theorem depends on definitions:  df-bi 116  df-clab 2157  df-cleq 2163  df-clel 2166  df-sbc 2956
This theorem is referenced by:  bdccsb  13895
  Copyright terms: Public domain W3C validator