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Theorem csbtt 3446
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  e.  V  /\  F/_ x B )  ->  [_ A  /  x ]_ B  =  B
)

Proof of Theorem csbtt
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 df-csb 3436 . 2  |-  [_ A  /  x ]_ B  =  { y  |  [. A  /  x ]. y  e.  B }
2 nfcr 2620 . . . 4  |-  ( F/_ x B  ->  F/ x  y  e.  B )
3 sbctt 3402 . . . 4  |-  ( ( A  e.  V  /\  F/ x  y  e.  B )  ->  ( [. A  /  x ]. y  e.  B  <->  y  e.  B ) )
42, 3sylan2 474 . . 3  |-  ( ( A  e.  V  /\  F/_ x B )  -> 
( [. A  /  x ]. y  e.  B  <->  y  e.  B ) )
54abbi1dv 2605 . 2  |-  ( ( A  e.  V  /\  F/_ x B )  ->  { y  |  [. A  /  x ]. y  e.  B }  =  B )
61, 5syl5eq 2520 1  |-  ( ( A  e.  V  /\  F/_ x B )  ->  [_ A  /  x ]_ B  =  B
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379   F/wnf 1599    e. wcel 1767   {cab 2452   F/_wnfc 2615   [.wsbc 3331   [_csb 3435
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-v 3115  df-sbc 3332  df-csb 3436
This theorem is referenced by:  csbconstgf  3447  sbnfc2  3854  constlimc  31194
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