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Theorem bds 9971
 Description: Boundedness of a formula resulting from implicit substitution in a bounded formula. Note that the proof does not use ax-bdsb 9942; therefore, using implicit instead of explicit substitution when boundedness is important, one might avoid using ax-bdsb 9942. (Contributed by BJ, 19-Nov-2019.)
Hypotheses
Ref Expression
bds.bd BOUNDED
bds.1
Assertion
Ref Expression
bds BOUNDED
Distinct variable groups:   ,   ,
Allowed substitution hints:   ()   ()

Proof of Theorem bds
StepHypRef Expression
1 bds.bd . . . 4 BOUNDED
21bdcab 9969 . . 3 BOUNDED
3 bds.1 . . . 4
43cbvabv 2161 . . 3
52, 4bdceqi 9963 . 2 BOUNDED
65bdph 9970 1 BOUNDED
 Colors of variables: wff set class Syntax hints:   wi 4   wb 98  cab 2026  BOUNDED wbd 9932 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-bd0 9933  ax-bdsb 9942 This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-bdc 9961 This theorem is referenced by: (None)
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