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Theorem bdsbc 10916
Description: A formula resulting from proper substitution of a setvar for a setvar in a bounded formula is bounded. See also bdsbcALT 10917. (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 10880 . 2  |- BOUNDED  [ y  /  x ] ph
3 sbsbc 2828 . 2  |-  ( [ y  /  x ] ph 
<-> 
[. y  /  x ]. ph )
42, 3bd0 10882 1  |- BOUNDED  [. y  /  x ]. ph
Colors of variables: wff set class
Syntax hints:   [wsb 1687   [.wsbc 2824  BOUNDED wbd 10870
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 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441  ax-17 1460  ax-ial 1468  ax-ext 2065  ax-bd0 10871  ax-bdsb 10880
This theorem depends on definitions:  df-bi 115  df-clab 2070  df-cleq 2076  df-clel 2079  df-sbc 2825
This theorem is referenced by:  bdccsb  10918
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