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Theorem csbtt 3009
 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 2999 . 2 𝐴 / 𝑥𝐵 = {𝑦[𝐴 / 𝑥]𝑦𝐵}
2 nfcr 2271 . . . 4 (𝑥𝐵 → Ⅎ𝑥 𝑦𝐵)
3 sbctt 2970 . . . 4 ((𝐴𝑉 ∧ Ⅎ𝑥 𝑦𝐵) → ([𝐴 / 𝑥]𝑦𝐵𝑦𝐵))
42, 3sylan2 284 . . 3 ((𝐴𝑉𝑥𝐵) → ([𝐴 / 𝑥]𝑦𝐵𝑦𝐵))
54abbi1dv 2257 . 2 ((𝐴𝑉𝑥𝐵) → {𝑦[𝐴 / 𝑥]𝑦𝐵} = 𝐵)
61, 5syl5eq 2182 1 ((𝐴𝑉𝑥𝐵) → 𝐴 / 𝑥𝐵 = 𝐵)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1331  Ⅎwnf 1436   ∈ wcel 1480  {cab 2123  Ⅎwnfc 2266  [wsbc 2904  ⦋csb 2998 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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119 This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-sbc 2905  df-csb 2999 This theorem is referenced by:  csbconstgf  3010  sbnfc2  3055
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