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Theorem csbtt 2856
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 ((A 𝑉 xB) → A / xB = B)

Proof of Theorem csbtt
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 df-csb 2847 . 2 A / xB = {y[A / x]y B}
2 nfcr 2167 . . . 4 (xB → Ⅎx y B)
3 sbctt 2818 . . . 4 ((A 𝑉 x y B) → ([A / x]y By B))
42, 3sylan2 270 . . 3 ((A 𝑉 xB) → ([A / x]y By B))
54abbi1dv 2154 . 2 ((A 𝑉 xB) → {y[A / x]y B} = B)
61, 5syl5eq 2081 1 ((A 𝑉 xB) → A / xB = B)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1242  wnf 1346   wcel 1390  {cab 2023  wnfc 2162  [wsbc 2758  csb 2846
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-sbc 2759  df-csb 2847
This theorem is referenced by:  csbconstgf  2857  sbnfc2  2900
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