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Theorem cbvreu 3086
Description: Change the bound variable of a restricted uniqueness 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 1683 . . . 4  |-  F/ z ( x  e.  A  /\  ph )
21sb8eu 2314 . . 3  |-  ( E! x ( x  e.  A  /\  ph )  <->  E! z [ z  /  x ] ( x  e.  A  /\  ph )
)
3 sban 2114 . . . 4  |-  ( [ z  /  x ]
( x  e.  A  /\  ph )  <->  ( [
z  /  x ]
x  e.  A  /\  [ z  /  x ] ph ) )
43eubii 2300 . . 3  |-  ( E! z [ z  /  x ] ( x  e.  A  /\  ph )  <->  E! z ( [ z  /  x ] x  e.  A  /\  [ z  /  x ] ph ) )
5 clelsb3 2588 . . . . . 6  |-  ( [ z  /  x ]
x  e.  A  <->  z  e.  A )
65anbi1i 695 . . . . 5  |-  ( ( [ z  /  x ] x  e.  A  /\  [ z  /  x ] ph )  <->  ( z  e.  A  /\  [ z  /  x ] ph ) )
76eubii 2300 . . . 4  |-  ( E! z ( [ z  /  x ] x  e.  A  /\  [ z  /  x ] ph ) 
<->  E! z ( z  e.  A  /\  [
z  /  x ] ph ) )
8 nfv 1683 . . . . . 6  |-  F/ y  z  e.  A
9 cbvral.1 . . . . . . 7  |-  F/ y
ph
109nfsb 2168 . . . . . 6  |-  F/ y [ z  /  x ] ph
118, 10nfan 1875 . . . . 5  |-  F/ y ( z  e.  A  /\  [ z  /  x ] ph )
12 nfv 1683 . . . . 5  |-  F/ z ( y  e.  A  /\  ps )
13 eleq1 2539 . . . . . 6  |-  ( z  =  y  ->  (
z  e.  A  <->  y  e.  A ) )
14 sbequ 2090 . . . . . . 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 2123 . . . . . . 7  |-  ( [ y  /  x ] ph 
<->  ps )
1814, 17syl6bb 261 . . . . . 6  |-  ( z  =  y  ->  ( [ z  /  x ] ph  <->  ps ) )
1913, 18anbi12d 710 . . . . 5  |-  ( z  =  y  ->  (
( z  e.  A  /\  [ z  /  x ] ph )  <->  ( y  e.  A  /\  ps )
) )
2011, 12, 19cbveu 2318 . . . 4  |-  ( E! z ( z  e.  A  /\  [ z  /  x ] ph ) 
<->  E! y ( y  e.  A  /\  ps ) )
217, 20bitri 249 . . 3  |-  ( E! z ( [ z  /  x ] x  e.  A  /\  [ z  /  x ] ph ) 
<->  E! y ( y  e.  A  /\  ps ) )
222, 4, 213bitri 271 . 2  |-  ( E! x ( x  e.  A  /\  ph )  <->  E! y ( y  e.  A  /\  ps )
)
23 df-reu 2821 . 2  |-  ( E! x  e.  A  ph  <->  E! x ( x  e.  A  /\  ph )
)
24 df-reu 2821 . 2  |-  ( E! y  e.  A  ps  <->  E! y ( y  e.  A  /\  ps )
)
2522, 23, 243bitr4i 277 1  |-  ( E! x  e.  A  ph  <->  E! y  e.  A  ps )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369   F/wnf 1599   [wsb 1711    e. wcel 1767   E!weu 2275   E!wreu 2816
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-cleq 2459  df-clel 2462  df-reu 2821
This theorem is referenced by:  cbvrmo  3087  cbvreuv  3090  fourierdlem50  31473
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