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Theorem csbtt 2941
Description: Substitution doesn't affect a constant 𝐵 (in which 𝑥 is not free). (Contributed by Mario Carneiro, 14-Oct-2016.)
Assertion
Ref Expression
csbtt ((𝐴𝑉𝑥𝐵) → 𝐴 / 𝑥𝐵 = 𝐵)

Proof of Theorem csbtt
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 df-csb 2932 . 2 𝐴 / 𝑥𝐵 = {𝑦[𝐴 / 𝑥]𝑦𝐵}
2 nfcr 2220 . . . 4 (𝑥𝐵 → Ⅎ𝑥 𝑦𝐵)
3 sbctt 2903 . . . 4 ((𝐴𝑉 ∧ Ⅎ𝑥 𝑦𝐵) → ([𝐴 / 𝑥]𝑦𝐵𝑦𝐵))
42, 3sylan2 280 . . 3 ((𝐴𝑉𝑥𝐵) → ([𝐴 / 𝑥]𝑦𝐵𝑦𝐵))
54abbi1dv 2207 . 2 ((𝐴𝑉𝑥𝐵) → {𝑦[𝐴 / 𝑥]𝑦𝐵} = 𝐵)
61, 5syl5eq 2132 1 ((𝐴𝑉𝑥𝐵) → 𝐴 / 𝑥𝐵 = 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103   = wceq 1289  wnf 1394  wcel 1438  {cab 2074  wnfc 2215  [wsbc 2838  csb 2931
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-sbc 2839  df-csb 2932
This theorem is referenced by:  csbconstgf  2942  sbnfc2  2986
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