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Theorem csbtt 3061
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 3050 . 2 𝐴 / 𝑥𝐵 = {𝑦[𝐴 / 𝑥]𝑦𝐵}
2 nfcr 2304 . . . 4 (𝑥𝐵 → Ⅎ𝑥 𝑦𝐵)
3 sbctt 3021 . . . 4 ((𝐴𝑉 ∧ Ⅎ𝑥 𝑦𝐵) → ([𝐴 / 𝑥]𝑦𝐵𝑦𝐵))
42, 3sylan2 284 . . 3 ((𝐴𝑉𝑥𝐵) → ([𝐴 / 𝑥]𝑦𝐵𝑦𝐵))
54abbi1dv 2290 . 2 ((𝐴𝑉𝑥𝐵) → {𝑦[𝐴 / 𝑥]𝑦𝐵} = 𝐵)
61, 5eqtrid 2215 1 ((𝐴𝑉𝑥𝐵) → 𝐴 / 𝑥𝐵 = 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1348  wnf 1453  wcel 2141  {cab 2156  wnfc 2299  [wsbc 2955  csb 3049
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-sbc 2956  df-csb 3050
This theorem is referenced by:  csbconstgf  3062  sbnfc2  3109
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