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Theorem bds 7078
Description: Boundedness of a formula resulting from implicit substitution in a bounded formula. Note that the proof does not use ax-bdsb 7049; therefore, using implicit instead of explicit substitution when boundedness is important, one might avoid using ax-bdsb 7049. (Contributed by BJ, 19-Nov-2019.)
Hypotheses
Ref Expression
bds.bd BOUNDED φ
bds.1 (x = y → (φψ))
Assertion
Ref Expression
bds BOUNDED ψ
Distinct variable groups:   ψ,x   φ,y
Allowed substitution hints:   φ(x)   ψ(y)

Proof of Theorem bds
StepHypRef Expression
1 bds.bd . . . 4 BOUNDED φ
21bdcab 7076 . . 3 BOUNDED {xφ}
3 bds.1 . . . 4 (x = y → (φψ))
43cbvabv 2143 . . 3 {xφ} = {yψ}
52, 4bdceqi 7070 . 2 BOUNDED {yψ}
65bdph 7077 1 BOUNDED ψ
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98  {cab 2008  BOUNDED wbd 7039
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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-bd0 7040  ax-bdsb 7049
This theorem depends on definitions:  df-bi 110  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-bdc 7068
This theorem is referenced by: (None)
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