Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bj-csbsnlem Structured version   Unicode version

Theorem bj-csbsnlem 34613
Description: Lemma for bj-csbsn 34614 (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 2444 . . . 4  |-  ( y  e.  { y  | 
[. A  /  x ]. y  e.  { x } }  <->  [. A  /  x ]. y  e.  { x } )
2 df-sbc 3328 . . . 4  |-  ( [. A  /  x ]. y  e.  { x }  <->  A  e.  { x  |  y  e. 
{ x } }
)
3 clelab 2601 . . . . 5  |-  ( A  e.  { x  |  y  e.  { x } }  <->  E. x ( x  =  A  /\  y  e.  { x } ) )
4 elsn 4046 . . . . . . 7  |-  ( y  e.  { x }  <->  y  =  x )
54anbi2i 694 . . . . . 6  |-  ( ( x  =  A  /\  y  e.  { x } )  <->  ( x  =  A  /\  y  =  x ) )
65exbii 1668 . . . . 5  |-  ( E. x ( x  =  A  /\  y  e. 
{ x } )  <->  E. x ( x  =  A  /\  y  =  x ) )
7 eqeq2 2472 . . . . . . . 8  |-  ( x  =  A  ->  (
y  =  x  <->  y  =  A ) )
87pm5.32i 637 . . . . . . 7  |-  ( ( x  =  A  /\  y  =  x )  <->  ( x  =  A  /\  y  =  A )
)
98exbii 1668 . . . . . 6  |-  ( E. x ( x  =  A  /\  y  =  x )  <->  E. x
( x  =  A  /\  y  =  A ) )
10 19.41v 1772 . . . . . 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 3117 . . . . . . . . 9  |-  ( y  =  A  ->  A  e.  _V )
13 elisset 3120 . . . . . . . . 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 3431 . . . 4  |-  [_ A  /  x ]_ { x }  =  { y  |  [. A  /  x ]. y  e.  { x } }
2120eleq2i 2535 . . 3  |-  ( y  e.  [_ A  /  x ]_ { x }  <->  y  e.  { y  | 
[. A  /  x ]. y  e.  { x } } )
22 elsn 4046 . . 3  |-  ( y  e.  { A }  <->  y  =  A )
2319, 21, 223bitr4i 277 . 2  |-  ( y  e.  [_ A  /  x ]_ { x }  <->  y  e.  { A }
)
2423eqriv 2453 1  |-  [_ A  /  x ]_ { x }  =  { A }
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    = wceq 1395   E.wex 1613    e. wcel 1819   {cab 2442   _Vcvv 3109   [.wsbc 3327   [_csb 3430   {csn 4032
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-v 3111  df-sbc 3328  df-csb 3431  df-sn 4033
This theorem is referenced by:  bj-csbsn  34614
  Copyright terms: Public domain W3C validator