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Theorem cbvrabcsf 3314
Description: A more general version of cbvrab 2954 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 1629 . . . 4  |-  F/ z ( x  e.  A  /\  ph )
2 nfcsb1v 3283 . . . . . 6  |-  F/_ x [_ z  /  x ]_ A
32nfcri 2566 . . . . 5  |-  F/ x  z  e.  [_ z  /  x ]_ A
4 nfs1v 2182 . . . . 5  |-  F/ x [ z  /  x ] ph
53, 4nfan 1846 . . . 4  |-  F/ x
( z  e.  [_ z  /  x ]_ A  /\  [ z  /  x ] ph )
6 id 20 . . . . . 6  |-  ( x  =  z  ->  x  =  z )
7 csbeq1a 3259 . . . . . 6  |-  ( x  =  z  ->  A  =  [_ z  /  x ]_ A )
86, 7eleq12d 2504 . . . . 5  |-  ( x  =  z  ->  (
x  e.  A  <->  z  e.  [_ z  /  x ]_ A ) )
9 sbequ12 1944 . . . . 5  |-  ( x  =  z  ->  ( ph 
<->  [ z  /  x ] ph ) )
108, 9anbi12d 692 . . . 4  |-  ( x  =  z  ->  (
( x  e.  A  /\  ph )  <->  ( z  e.  [_ z  /  x ]_ A  /\  [ z  /  x ] ph ) ) )
111, 5, 10cbvab 2554 . . 3  |-  { x  |  ( x  e.  A  /\  ph ) }  =  { z  |  ( z  e. 
[_ z  /  x ]_ A  /\  [ z  /  x ] ph ) }
12 nfcv 2572 . . . . . . 7  |-  F/_ y
z
13 cbvralcsf.1 . . . . . . 7  |-  F/_ y A
1412, 13nfcsb 3285 . . . . . 6  |-  F/_ y [_ z  /  x ]_ A
1514nfcri 2566 . . . . 5  |-  F/ y  z  e.  [_ z  /  x ]_ A
16 cbvralcsf.3 . . . . . 6  |-  F/ y
ph
1716nfsb 2185 . . . . 5  |-  F/ y [ z  /  x ] ph
1815, 17nfan 1846 . . . 4  |-  F/ y ( z  e.  [_ z  /  x ]_ A  /\  [ z  /  x ] ph )
19 nfv 1629 . . . 4  |-  F/ z ( y  e.  B  /\  ps )
20 id 20 . . . . . 6  |-  ( z  =  y  ->  z  =  y )
21 csbeq1 3254 . . . . . . 7  |-  ( z  =  y  ->  [_ z  /  x ]_ A  = 
[_ y  /  x ]_ A )
22 df-csb 3252 . . . . . . . 8  |-  [_ y  /  x ]_ A  =  { v  |  [. y  /  x ]. v  e.  A }
23 cbvralcsf.2 . . . . . . . . . . . 12  |-  F/_ x B
2423nfcri 2566 . . . . . . . . . . 11  |-  F/ x  v  e.  B
25 cbvralcsf.5 . . . . . . . . . . . 12  |-  ( x  =  y  ->  A  =  B )
2625eleq2d 2503 . . . . . . . . . . 11  |-  ( x  =  y  ->  (
v  e.  A  <->  v  e.  B ) )
2724, 26sbie 2149 . . . . . . . . . 10  |-  ( [ y  /  x ]
v  e.  A  <->  v  e.  B )
28 sbsbc 3165 . . . . . . . . . 10  |-  ( [ y  /  x ]
v  e.  A  <->  [. y  /  x ]. v  e.  A
)
2927, 28bitr3i 243 . . . . . . . . 9  |-  ( v  e.  B  <->  [. y  /  x ]. v  e.  A
)
3029abbi2i 2547 . . . . . . . 8  |-  B  =  { v  |  [. y  /  x ]. v  e.  A }
3122, 30eqtr4i 2459 . . . . . . 7  |-  [_ y  /  x ]_ A  =  B
3221, 31syl6eq 2484 . . . . . 6  |-  ( z  =  y  ->  [_ z  /  x ]_ A  =  B )
3320, 32eleq12d 2504 . . . . 5  |-  ( z  =  y  ->  (
z  e.  [_ z  /  x ]_ A  <->  y  e.  B ) )
34 sbequ 2111 . . . . . 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 2149 . . . . . 6  |-  ( [ y  /  x ] ph 
<->  ps )
3834, 37syl6bb 253 . . . . 5  |-  ( z  =  y  ->  ( [ z  /  x ] ph  <->  ps ) )
3933, 38anbi12d 692 . . . 4  |-  ( z  =  y  ->  (
( z  e.  [_ z  /  x ]_ A  /\  [ z  /  x ] ph )  <->  ( y  e.  B  /\  ps )
) )
4018, 19, 39cbvab 2554 . . 3  |-  { z  |  ( z  e. 
[_ z  /  x ]_ A  /\  [ z  /  x ] ph ) }  =  {
y  |  ( y  e.  B  /\  ps ) }
4111, 40eqtri 2456 . 2  |-  { x  |  ( x  e.  A  /\  ph ) }  =  { y  |  ( y  e.  B  /\  ps ) }
42 df-rab 2714 . 2  |-  { x  e.  A  |  ph }  =  { x  |  ( x  e.  A  /\  ph ) }
43 df-rab 2714 . 2  |-  { y  e.  B  |  ps }  =  { y  |  ( y  e.  B  /\  ps ) }
4441, 42, 433eqtr4i 2466 1  |-  { x  e.  A  |  ph }  =  { y  e.  B  |  ps }
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   F/wnf 1553    = wceq 1652   [wsb 1658    e. wcel 1725   {cab 2422   F/_wnfc 2559   {crab 2709   [.wsbc 3161   [_csb 3251
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-rab 2714  df-sbc 3162  df-csb 3252
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