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Theorem bdsbc 13853
Description: A formula resulting from proper substitution of a setvar for a setvar in a bounded formula is bounded. See also bdsbcALT 13854. (Contributed by BJ, 16-Oct-2019.)
Hypothesis
Ref Expression
bdcsbc.1  |- BOUNDED  ph
Assertion
Ref Expression
bdsbc  |- BOUNDED  [. y  /  x ]. ph

Proof of Theorem bdsbc
StepHypRef Expression
1 bdcsbc.1 . . 3  |- BOUNDED  ph
21ax-bdsb 13817 . 2  |- BOUNDED  [ y  /  x ] ph
3 sbsbc 2959 . 2  |-  ( [ y  /  x ] ph 
<-> 
[. y  /  x ]. ph )
42, 3bd0 13819 1  |- BOUNDED  [. y  /  x ]. ph
Colors of variables: wff set class
Syntax hints:   [wsb 1755   [.wsbc 2955  BOUNDED wbd 13807
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 13808  ax-bdsb 13817
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  13855
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