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Theorem csbtt 2835
 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 2826 . 2 A / xB = {y[A / x]y B}
2 nfcr 2148 . . . 4 (xB → Ⅎx y B)
3 sbctt 2797 . . . 4 ((A 𝑉 x y B) → ([A / x]y By B))
42, 3sylan2 270 . . 3 ((A 𝑉 xB) → ([A / x]y By B))
54abbi1dv 2135 . 2 ((A 𝑉 xB) → {y[A / x]y B} = B)
61, 5syl5eq 2062 1 ((A 𝑉 xB) → A / xB = B)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1226  Ⅎwnf 1325   ∈ wcel 1370  {cab 2004  Ⅎwnfc 2143  [wsbc 2737  ⦋csb 2825 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 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000 This theorem depends on definitions:  df-bi 110  df-tru 1229  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-v 2533  df-sbc 2738  df-csb 2826 This theorem is referenced by:  csbconstgf  2836  sbnfc2  2879
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