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Theorem csbtt 3127
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 3116 . 2  |-  [_ A  /  x ]_ B  =  { y  |  [. A  /  x ]. y  e.  B }
2 nfcr 2444 . . . 4  |-  ( F/_ x B  ->  F/ x  y  e.  B )
3 sbctt 3087 . . . 4  |-  ( ( A  e.  V  /\  F/ x  y  e.  B )  ->  ( [. A  /  x ]. y  e.  B  <->  y  e.  B ) )
42, 3sylan2 460 . . 3  |-  ( ( A  e.  V  /\  F/_ x B )  -> 
( [. A  /  x ]. y  e.  B  <->  y  e.  B ) )
54abbi1dv 2432 . 2  |-  ( ( A  e.  V  /\  F/_ x B )  ->  { y  |  [. A  /  x ]. y  e.  B }  =  B )
61, 5syl5eq 2360 1  |-  ( ( A  e.  V  /\  F/_ x B )  ->  [_ A  /  x ]_ B  =  B
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   F/wnf 1535    = wceq 1633    e. wcel 1701   {cab 2302   F/_wnfc 2439   [.wsbc 3025   [_csb 3115
This theorem is referenced by:  csbconstgf  3128  sbnfc2  3175
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-v 2824  df-sbc 3026  df-csb 3116
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