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Theorem csbtt 3043
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 3032 . 2 𝐴 / 𝑥𝐵 = {𝑦[𝐴 / 𝑥]𝑦𝐵}
2 nfcr 2291 . . . 4 (𝑥𝐵 → Ⅎ𝑥 𝑦𝐵)
3 sbctt 3003 . . . 4 ((𝐴𝑉 ∧ Ⅎ𝑥 𝑦𝐵) → ([𝐴 / 𝑥]𝑦𝐵𝑦𝐵))
42, 3sylan2 284 . . 3 ((𝐴𝑉𝑥𝐵) → ([𝐴 / 𝑥]𝑦𝐵𝑦𝐵))
54abbi1dv 2277 . 2 ((𝐴𝑉𝑥𝐵) → {𝑦[𝐴 / 𝑥]𝑦𝐵} = 𝐵)
61, 5syl5eq 2202 1 ((𝐴𝑉𝑥𝐵) → 𝐴 / 𝑥𝐵 = 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1335  wnf 1440  wcel 2128  {cab 2143  wnfc 2286  [wsbc 2937  csb 3031
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-sbc 2938  df-csb 3032
This theorem is referenced by:  csbconstgf  3044  sbnfc2  3091
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