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Theorem csbtt 3148
 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 / x]B = B)

Proof of Theorem csbtt
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 df-csb 3137 . 2 [A / x]B = {y A / xy B}
2 nfcr 2481 . . . 4 (xB → Ⅎx y B)
3 sbctt 3108 . . . 4 ((A V x y B) → ([̣A / xy By B))
42, 3sylan2 460 . . 3 ((A V xB) → ([̣A / xy By B))
54abbi1dv 2469 . 2 ((A V xB) → {y A / xy B} = B)
61, 5syl5eq 2397 1 ((A V xB) → [A / x]B = B)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  Ⅎwnf 1544   = wceq 1642   ∈ wcel 1710  {cab 2339  Ⅎwnfc 2476  [̣wsbc 3046  [csb 3136 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-sbc 3047  df-csb 3137 This theorem is referenced by:  csbconstgf  3149  sbnfc2  3196
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