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Theorem bj-csbsnlem 32702
Description: Lemma for bj-csbsn 32703 (in this lemma,  x cannot occur in  A). (Contributed by BJ, 6-Oct-2018.) (New usage is discouraged.)
Assertion
Ref Expression
bj-csbsnlem  |-  [_ A  /  x ]_ { x }  =  { A }
Distinct variable group:    x, A

Proof of Theorem bj-csbsnlem
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 abid 2438 . . . 4  |-  ( y  e.  { y  | 
[. A  /  x ]. y  e.  { x } }  <->  [. A  /  x ]. y  e.  { x } )
2 df-sbc 3282 . . . 4  |-  ( [. A  /  x ]. y  e.  { x }  <->  A  e.  { x  |  y  e. 
{ x } }
)
3 clelab 2593 . . . . 5  |-  ( A  e.  { x  |  y  e.  { x } }  <->  E. x ( x  =  A  /\  y  e.  { x } ) )
4 elsn 3986 . . . . . . 7  |-  ( y  e.  { x }  <->  y  =  x )
54anbi2i 694 . . . . . 6  |-  ( ( x  =  A  /\  y  e.  { x } )  <->  ( x  =  A  /\  y  =  x ) )
65exbii 1635 . . . . 5  |-  ( E. x ( x  =  A  /\  y  e. 
{ x } )  <->  E. x ( x  =  A  /\  y  =  x ) )
7 eqeq2 2465 . . . . . . . 8  |-  ( x  =  A  ->  (
y  =  x  <->  y  =  A ) )
87pm5.32i 637 . . . . . . 7  |-  ( ( x  =  A  /\  y  =  x )  <->  ( x  =  A  /\  y  =  A )
)
98exbii 1635 . . . . . 6  |-  ( E. x ( x  =  A  /\  y  =  x )  <->  E. x
( x  =  A  /\  y  =  A ) )
10 19.41v 1929 . . . . . 6  |-  ( E. x ( x  =  A  /\  y  =  A )  <->  ( E. x  x  =  A  /\  y  =  A
) )
11 simpr 461 . . . . . . 7  |-  ( ( E. x  x  =  A  /\  y  =  A )  ->  y  =  A )
12 eqvisset 3073 . . . . . . . . 9  |-  ( y  =  A  ->  A  e.  _V )
13 elisset 3076 . . . . . . . . 9  |-  ( A  e.  _V  ->  E. x  x  =  A )
1412, 13syl 16 . . . . . . . 8  |-  ( y  =  A  ->  E. x  x  =  A )
1514ancri 552 . . . . . . 7  |-  ( y  =  A  ->  ( E. x  x  =  A  /\  y  =  A ) )
1611, 15impbii 188 . . . . . 6  |-  ( ( E. x  x  =  A  /\  y  =  A )  <->  y  =  A )
179, 10, 163bitri 271 . . . . 5  |-  ( E. x ( x  =  A  /\  y  =  x )  <->  y  =  A )
183, 6, 173bitri 271 . . . 4  |-  ( A  e.  { x  |  y  e.  { x } }  <->  y  =  A )
191, 2, 183bitri 271 . . 3  |-  ( y  e.  { y  | 
[. A  /  x ]. y  e.  { x } }  <->  y  =  A )
20 df-csb 3384 . . . 4  |-  [_ A  /  x ]_ { x }  =  { y  |  [. A  /  x ]. y  e.  { x } }
2120eleq2i 2527 . . 3  |-  ( y  e.  [_ A  /  x ]_ { x }  <->  y  e.  { y  | 
[. A  /  x ]. y  e.  { x } } )
22 elsn 3986 . . 3  |-  ( y  e.  { A }  <->  y  =  A )
2319, 21, 223bitr4i 277 . 2  |-  ( y  e.  [_ A  /  x ]_ { x }  <->  y  e.  { A }
)
2423eqriv 2447 1  |-  [_ A  /  x ]_ { x }  =  { A }
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    = wceq 1370   E.wex 1587    e. wcel 1758   {cab 2436   _Vcvv 3065   [.wsbc 3281   [_csb 3383   {csn 3972
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-v 3067  df-sbc 3282  df-csb 3384  df-sn 3973
This theorem is referenced by:  bj-csbsn  32703
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