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Theorem cbvrabcsf 3159
Description: A more general version of cbvrab 2799 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 1609 . . . 4  |-  F/ z ( x  e.  A  /\  ph )
2 nfcsb1v 3126 . . . . . 6  |-  F/_ x [_ z  /  x ]_ A
32nfcri 2426 . . . . 5  |-  F/ x  z  e.  [_ z  /  x ]_ A
4 nfs1v 2058 . . . . 5  |-  F/ x [ z  /  x ] ph
53, 4nfan 1783 . . . 4  |-  F/ x
( z  e.  [_ z  /  x ]_ A  /\  [ z  /  x ] ph )
6 id 19 . . . . . 6  |-  ( x  =  z  ->  x  =  z )
7 csbeq1a 3102 . . . . . 6  |-  ( x  =  z  ->  A  =  [_ z  /  x ]_ A )
86, 7eleq12d 2364 . . . . 5  |-  ( x  =  z  ->  (
x  e.  A  <->  z  e.  [_ z  /  x ]_ A ) )
9 sbequ12 1872 . . . . 5  |-  ( x  =  z  ->  ( ph 
<->  [ z  /  x ] ph ) )
108, 9anbi12d 691 . . . 4  |-  ( x  =  z  ->  (
( x  e.  A  /\  ph )  <->  ( z  e.  [_ z  /  x ]_ A  /\  [ z  /  x ] ph ) ) )
111, 5, 10cbvab 2414 . . 3  |-  { x  |  ( x  e.  A  /\  ph ) }  =  { z  |  ( z  e. 
[_ z  /  x ]_ A  /\  [ z  /  x ] ph ) }
12 nfcv 2432 . . . . . . 7  |-  F/_ y
z
13 cbvralcsf.1 . . . . . . 7  |-  F/_ y A
1412, 13nfcsb 3128 . . . . . 6  |-  F/_ y [_ z  /  x ]_ A
1514nfcri 2426 . . . . 5  |-  F/ y  z  e.  [_ z  /  x ]_ A
16 cbvralcsf.3 . . . . . 6  |-  F/ y
ph
1716nfsb 2061 . . . . 5  |-  F/ y [ z  /  x ] ph
1815, 17nfan 1783 . . . 4  |-  F/ y ( z  e.  [_ z  /  x ]_ A  /\  [ z  /  x ] ph )
19 nfv 1609 . . . 4  |-  F/ z ( y  e.  B  /\  ps )
20 id 19 . . . . . 6  |-  ( z  =  y  ->  z  =  y )
21 csbeq1 3097 . . . . . . 7  |-  ( z  =  y  ->  [_ z  /  x ]_ A  = 
[_ y  /  x ]_ A )
22 df-csb 3095 . . . . . . . 8  |-  [_ y  /  x ]_ A  =  { v  |  [. y  /  x ]. v  e.  A }
23 cbvralcsf.2 . . . . . . . . . . . 12  |-  F/_ x B
2423nfcri 2426 . . . . . . . . . . 11  |-  F/ x  v  e.  B
25 cbvralcsf.5 . . . . . . . . . . . 12  |-  ( x  =  y  ->  A  =  B )
2625eleq2d 2363 . . . . . . . . . . 11  |-  ( x  =  y  ->  (
v  e.  A  <->  v  e.  B ) )
2724, 26sbie 1991 . . . . . . . . . 10  |-  ( [ y  /  x ]
v  e.  A  <->  v  e.  B )
28 sbsbc 3008 . . . . . . . . . 10  |-  ( [ y  /  x ]
v  e.  A  <->  [. y  /  x ]. v  e.  A
)
2927, 28bitr3i 242 . . . . . . . . 9  |-  ( v  e.  B  <->  [. y  /  x ]. v  e.  A
)
3029abbi2i 2407 . . . . . . . 8  |-  B  =  { v  |  [. y  /  x ]. v  e.  A }
3122, 30eqtr4i 2319 . . . . . . 7  |-  [_ y  /  x ]_ A  =  B
3221, 31syl6eq 2344 . . . . . 6  |-  ( z  =  y  ->  [_ z  /  x ]_ A  =  B )
3320, 32eleq12d 2364 . . . . 5  |-  ( z  =  y  ->  (
z  e.  [_ z  /  x ]_ A  <->  y  e.  B ) )
34 sbequ 2013 . . . . . 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 1991 . . . . . 6  |-  ( [ y  /  x ] ph 
<->  ps )
3834, 37syl6bb 252 . . . . 5  |-  ( z  =  y  ->  ( [ z  /  x ] ph  <->  ps ) )
3933, 38anbi12d 691 . . . 4  |-  ( z  =  y  ->  (
( z  e.  [_ z  /  x ]_ A  /\  [ z  /  x ] ph )  <->  ( y  e.  B  /\  ps )
) )
4018, 19, 39cbvab 2414 . . 3  |-  { z  |  ( z  e. 
[_ z  /  x ]_ A  /\  [ z  /  x ] ph ) }  =  {
y  |  ( y  e.  B  /\  ps ) }
4111, 40eqtri 2316 . 2  |-  { x  |  ( x  e.  A  /\  ph ) }  =  { y  |  ( y  e.  B  /\  ps ) }
42 df-rab 2565 . 2  |-  { x  e.  A  |  ph }  =  { x  |  ( x  e.  A  /\  ph ) }
43 df-rab 2565 . 2  |-  { y  e.  B  |  ps }  =  { y  |  ( y  e.  B  /\  ps ) }
4441, 42, 433eqtr4i 2326 1  |-  { x  e.  A  |  ph }  =  { y  e.  B  |  ps }
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   F/wnf 1534    = wceq 1632   [wsb 1638    e. wcel 1696   {cab 2282   F/_wnfc 2419   {crab 2560   [.wsbc 3004   [_csb 3094
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rab 2565  df-sbc 3005  df-csb 3095
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