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Theorem cbvrabcsf 3410
Description: A more general version of cbvrab 3059 with no distinct variable restrictions. (Contributed by Andrew Salmon, 13-Jul-2011.)
Hypotheses
Ref Expression
cbvralcsf.1  |-  F/_ y A
cbvralcsf.2  |-  F/_ x B
cbvralcsf.3  |-  F/ y
ph
cbvralcsf.4  |-  F/ x ps
cbvralcsf.5  |-  ( x  =  y  ->  A  =  B )
cbvralcsf.6  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
cbvrabcsf  |-  { x  e.  A  |  ph }  =  { y  e.  B  |  ps }

Proof of Theorem cbvrabcsf
Dummy variables  v 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfv 1730 . . . 4  |-  F/ z ( x  e.  A  /\  ph )
2 nfcsb1v 3391 . . . . . 6  |-  F/_ x [_ z  /  x ]_ A
32nfcri 2559 . . . . 5  |-  F/ x  z  e.  [_ z  /  x ]_ A
4 nfs1v 2207 . . . . 5  |-  F/ x [ z  /  x ] ph
53, 4nfan 1958 . . . 4  |-  F/ x
( z  e.  [_ z  /  x ]_ A  /\  [ z  /  x ] ph )
6 id 23 . . . . . 6  |-  ( x  =  z  ->  x  =  z )
7 csbeq1a 3384 . . . . . 6  |-  ( x  =  z  ->  A  =  [_ z  /  x ]_ A )
86, 7eleq12d 2486 . . . . 5  |-  ( x  =  z  ->  (
x  e.  A  <->  z  e.  [_ z  /  x ]_ A ) )
9 sbequ12 2022 . . . . 5  |-  ( x  =  z  ->  ( ph 
<->  [ z  /  x ] ph ) )
108, 9anbi12d 711 . . . 4  |-  ( x  =  z  ->  (
( x  e.  A  /\  ph )  <->  ( z  e.  [_ z  /  x ]_ A  /\  [ z  /  x ] ph ) ) )
111, 5, 10cbvab 2545 . . 3  |-  { x  |  ( x  e.  A  /\  ph ) }  =  { z  |  ( z  e. 
[_ z  /  x ]_ A  /\  [ z  /  x ] ph ) }
12 nfcv 2566 . . . . . . 7  |-  F/_ y
z
13 cbvralcsf.1 . . . . . . 7  |-  F/_ y A
1412, 13nfcsb 3393 . . . . . 6  |-  F/_ y [_ z  /  x ]_ A
1514nfcri 2559 . . . . 5  |-  F/ y  z  e.  [_ z  /  x ]_ A
16 cbvralcsf.3 . . . . . 6  |-  F/ y
ph
1716nfsb 2210 . . . . 5  |-  F/ y [ z  /  x ] ph
1815, 17nfan 1958 . . . 4  |-  F/ y ( z  e.  [_ z  /  x ]_ A  /\  [ z  /  x ] ph )
19 nfv 1730 . . . 4  |-  F/ z ( y  e.  B  /\  ps )
20 id 23 . . . . . 6  |-  ( z  =  y  ->  z  =  y )
21 csbeq1 3378 . . . . . . 7  |-  ( z  =  y  ->  [_ z  /  x ]_ A  = 
[_ y  /  x ]_ A )
22 df-csb 3376 . . . . . . . 8  |-  [_ y  /  x ]_ A  =  { v  |  [. y  /  x ]. v  e.  A }
23 cbvralcsf.2 . . . . . . . . . . . 12  |-  F/_ x B
2423nfcri 2559 . . . . . . . . . . 11  |-  F/ x  v  e.  B
25 cbvralcsf.5 . . . . . . . . . . . 12  |-  ( x  =  y  ->  A  =  B )
2625eleq2d 2474 . . . . . . . . . . 11  |-  ( x  =  y  ->  (
v  e.  A  <->  v  e.  B ) )
2724, 26sbie 2175 . . . . . . . . . 10  |-  ( [ y  /  x ]
v  e.  A  <->  v  e.  B )
28 sbsbc 3283 . . . . . . . . . 10  |-  ( [ y  /  x ]
v  e.  A  <->  [. y  /  x ]. v  e.  A
)
2927, 28bitr3i 253 . . . . . . . . 9  |-  ( v  e.  B  <->  [. y  /  x ]. v  e.  A
)
3029abbi2i 2537 . . . . . . . 8  |-  B  =  { v  |  [. y  /  x ]. v  e.  A }
3122, 30eqtr4i 2436 . . . . . . 7  |-  [_ y  /  x ]_ A  =  B
3221, 31syl6eq 2461 . . . . . 6  |-  ( z  =  y  ->  [_ z  /  x ]_ A  =  B )
3320, 32eleq12d 2486 . . . . 5  |-  ( z  =  y  ->  (
z  e.  [_ z  /  x ]_ A  <->  y  e.  B ) )
34 sbequ 2143 . . . . . 6  |-  ( z  =  y  ->  ( [ z  /  x ] ph  <->  [ y  /  x ] ph ) )
35 cbvralcsf.4 . . . . . . 7  |-  F/ x ps
36 cbvralcsf.6 . . . . . . 7  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
3735, 36sbie 2175 . . . . . 6  |-  ( [ y  /  x ] ph 
<->  ps )
3834, 37syl6bb 263 . . . . 5  |-  ( z  =  y  ->  ( [ z  /  x ] ph  <->  ps ) )
3933, 38anbi12d 711 . . . 4  |-  ( z  =  y  ->  (
( z  e.  [_ z  /  x ]_ A  /\  [ z  /  x ] ph )  <->  ( y  e.  B  /\  ps )
) )
4018, 19, 39cbvab 2545 . . 3  |-  { z  |  ( z  e. 
[_ z  /  x ]_ A  /\  [ z  /  x ] ph ) }  =  {
y  |  ( y  e.  B  /\  ps ) }
4111, 40eqtri 2433 . 2  |-  { x  |  ( x  e.  A  /\  ph ) }  =  { y  |  ( y  e.  B  /\  ps ) }
42 df-rab 2765 . 2  |-  { x  e.  A  |  ph }  =  { x  |  ( x  e.  A  /\  ph ) }
43 df-rab 2765 . 2  |-  { y  e.  B  |  ps }  =  { y  |  ( y  e.  B  /\  ps ) }
4441, 42, 433eqtr4i 2443 1  |-  { x  e.  A  |  ph }  =  { y  e.  B  |  ps }
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 186    /\ wa 369    = wceq 1407   F/wnf 1639   [wsb 1765    e. wcel 1844   {cab 2389   F/_wnfc 2552   {crab 2760   [.wsbc 3279   [_csb 3375
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-rab 2765  df-sbc 3280  df-csb 3376
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator