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Theorem bds 10909
 Description: Boundedness of a formula resulting from implicit substitution in a bounded formula. Note that the proof does not use ax-bdsb 10880; therefore, using implicit instead of explicit substitution when boundedness is important, one might avoid using ax-bdsb 10880. (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 10907 . . 3 BOUNDED {𝑥𝜑}
3 bds.1 . . . 4 (𝑥 = 𝑦 → (𝜑𝜓))
43cbvabv 2206 . . 3 {𝑥𝜑} = {𝑦𝜓}
52, 4bdceqi 10901 . 2 BOUNDED {𝑦𝜓}
65bdph 10908 1 BOUNDED 𝜓
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 103  {cab 2069  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-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-bd0 10871  ax-bdsb 10880 This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-bdc 10899 This theorem is referenced by: (None)
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