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Theorem cbvreu 2588
Description: Change the bound variable of a restricted unique existential quantifier using implicit substitution. (Contributed by Mario Carneiro, 15-Oct-2016.)
Hypotheses
Ref Expression
cbvral.1  |-  F/ y
ph
cbvral.2  |-  F/ x ps
cbvral.3  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
cbvreu  |-  ( E! x  e.  A  ph  <->  E! y  e.  A  ps )
Distinct variable groups:    x, A    y, A
Allowed substitution hints:    ph( x, y)    ps( x, y)

Proof of Theorem cbvreu
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 nfv 1466 . . . 4  |-  F/ z ( x  e.  A  /\  ph )
21sb8eu 1961 . . 3  |-  ( E! x ( x  e.  A  /\  ph )  <->  E! z [ z  /  x ] ( x  e.  A  /\  ph )
)
3 sban 1877 . . . 4  |-  ( [ z  /  x ]
( x  e.  A  /\  ph )  <->  ( [
z  /  x ]
x  e.  A  /\  [ z  /  x ] ph ) )
43eubii 1957 . . 3  |-  ( E! z [ z  /  x ] ( x  e.  A  /\  ph )  <->  E! z ( [ z  /  x ] x  e.  A  /\  [ z  /  x ] ph ) )
5 clelsb3 2192 . . . . . 6  |-  ( [ z  /  x ]
x  e.  A  <->  z  e.  A )
65anbi1i 446 . . . . 5  |-  ( ( [ z  /  x ] x  e.  A  /\  [ z  /  x ] ph )  <->  ( z  e.  A  /\  [ z  /  x ] ph ) )
76eubii 1957 . . . 4  |-  ( E! z ( [ z  /  x ] x  e.  A  /\  [ z  /  x ] ph ) 
<->  E! z ( z  e.  A  /\  [
z  /  x ] ph ) )
8 nfv 1466 . . . . . 6  |-  F/ y  z  e.  A
9 cbvral.1 . . . . . . 7  |-  F/ y
ph
109nfsb 1870 . . . . . 6  |-  F/ y [ z  /  x ] ph
118, 10nfan 1502 . . . . 5  |-  F/ y ( z  e.  A  /\  [ z  /  x ] ph )
12 nfv 1466 . . . . 5  |-  F/ z ( y  e.  A  /\  ps )
13 eleq1 2150 . . . . . 6  |-  ( z  =  y  ->  (
z  e.  A  <->  y  e.  A ) )
14 sbequ 1768 . . . . . . 7  |-  ( z  =  y  ->  ( [ z  /  x ] ph  <->  [ y  /  x ] ph ) )
15 cbvral.2 . . . . . . . 8  |-  F/ x ps
16 cbvral.3 . . . . . . . 8  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
1715, 16sbie 1721 . . . . . . 7  |-  ( [ y  /  x ] ph 
<->  ps )
1814, 17syl6bb 194 . . . . . 6  |-  ( z  =  y  ->  ( [ z  /  x ] ph  <->  ps ) )
1913, 18anbi12d 457 . . . . 5  |-  ( z  =  y  ->  (
( z  e.  A  /\  [ z  /  x ] ph )  <->  ( y  e.  A  /\  ps )
) )
2011, 12, 19cbveu 1972 . . . 4  |-  ( E! z ( z  e.  A  /\  [ z  /  x ] ph ) 
<->  E! y ( y  e.  A  /\  ps ) )
217, 20bitri 182 . . 3  |-  ( E! z ( [ z  /  x ] x  e.  A  /\  [ z  /  x ] ph ) 
<->  E! y ( y  e.  A  /\  ps ) )
222, 4, 213bitri 204 . 2  |-  ( E! x ( x  e.  A  /\  ph )  <->  E! y ( y  e.  A  /\  ps )
)
23 df-reu 2366 . 2  |-  ( E! x  e.  A  ph  <->  E! x ( x  e.  A  /\  ph )
)
24 df-reu 2366 . 2  |-  ( E! y  e.  A  ps  <->  E! y ( y  e.  A  /\  ps )
)
2522, 23, 243bitr4i 210 1  |-  ( E! x  e.  A  ph  <->  E! y  e.  A  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103   F/wnf 1394    e. wcel 1438   [wsb 1692   E!weu 1948   E!wreu 2361
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-cleq 2081  df-clel 2084  df-reu 2366
This theorem is referenced by:  cbvrmo  2589  cbvreuv  2592
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