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Theorem rexxpf 4511
Description: Version of rexxp 4508 with bound-variable hypotheses. (Contributed by NM, 19-Dec-2008.) (Revised by Mario Carneiro, 15-Oct-2016.)
Hypotheses
Ref Expression
ralxpf.1  |-  F/ y
ph
ralxpf.2  |-  F/ z
ph
ralxpf.3  |-  F/ x ps
ralxpf.4  |-  ( x  =  <. y ,  z
>.  ->  ( ph  <->  ps )
)
Assertion
Ref Expression
rexxpf  |-  ( E. x  e.  ( A  X.  B ) ph  <->  E. y  e.  A  E. z  e.  B  ps )
Distinct variable groups:    x, y, A   
x, z, B, y
Allowed substitution hints:    ph( x, y, z)    ps( x, y, z)    A( z)

Proof of Theorem rexxpf
Dummy variables  v  u  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cbvrexsv 2590 . 2  |-  ( E. x  e.  ( A  X.  B ) ph  <->  E. v  e.  ( A  X.  B ) [ v  /  x ] ph )
2 cbvrexsv 2590 . . . 4  |-  ( E. z  e.  B  [
w  /  y ] ps  <->  E. u  e.  B  [ u  /  z ] [ w  /  y ] ps )
32rexbii 2374 . . 3  |-  ( E. w  e.  A  E. z  e.  B  [
w  /  y ] ps  <->  E. w  e.  A  E. u  e.  B  [ u  /  z ] [ w  /  y ] ps )
4 nfv 1462 . . . 4  |-  F/ w E. z  e.  B  ps
5 nfcv 2220 . . . . 5  |-  F/_ y B
6 nfs1v 1857 . . . . 5  |-  F/ y [ w  /  y ] ps
75, 6nfrexxy 2404 . . . 4  |-  F/ y E. z  e.  B  [ w  /  y ] ps
8 sbequ12 1695 . . . . 5  |-  ( y  =  w  ->  ( ps 
<->  [ w  /  y ] ps ) )
98rexbidv 2370 . . . 4  |-  ( y  =  w  ->  ( E. z  e.  B  ps 
<->  E. z  e.  B  [ w  /  y ] ps ) )
104, 7, 9cbvrex 2575 . . 3  |-  ( E. y  e.  A  E. z  e.  B  ps  <->  E. w  e.  A  E. z  e.  B  [
w  /  y ] ps )
11 vex 2605 . . . . . 6  |-  w  e. 
_V
12 vex 2605 . . . . . 6  |-  u  e. 
_V
1311, 12eqvinop 4006 . . . . 5  |-  ( v  =  <. w ,  u >.  <->  E. y E. z ( v  =  <. y ,  z >.  /\  <. y ,  z >.  =  <. w ,  u >. )
)
14 ralxpf.1 . . . . . . . 8  |-  F/ y
ph
1514nfsb 1864 . . . . . . 7  |-  F/ y [ v  /  x ] ph
166nfsb 1864 . . . . . . 7  |-  F/ y [ u  /  z ] [ w  /  y ] ps
1715, 16nfbi 1522 . . . . . 6  |-  F/ y ( [ v  /  x ] ph  <->  [ u  /  z ] [
w  /  y ] ps )
18 ralxpf.2 . . . . . . . . 9  |-  F/ z
ph
1918nfsb 1864 . . . . . . . 8  |-  F/ z [ v  /  x ] ph
20 nfs1v 1857 . . . . . . . 8  |-  F/ z [ u  /  z ] [ w  /  y ] ps
2119, 20nfbi 1522 . . . . . . 7  |-  F/ z ( [ v  /  x ] ph  <->  [ u  /  z ] [
w  /  y ] ps )
22 ralxpf.3 . . . . . . . . 9  |-  F/ x ps
23 ralxpf.4 . . . . . . . . 9  |-  ( x  =  <. y ,  z
>.  ->  ( ph  <->  ps )
)
2422, 23sbhypf 2649 . . . . . . . 8  |-  ( v  =  <. y ,  z
>.  ->  ( [ v  /  x ] ph  <->  ps ) )
25 vex 2605 . . . . . . . . . 10  |-  y  e. 
_V
26 vex 2605 . . . . . . . . . 10  |-  z  e. 
_V
2725, 26opth 4000 . . . . . . . . 9  |-  ( <.
y ,  z >.  =  <. w ,  u >.  <-> 
( y  =  w  /\  z  =  u ) )
28 sbequ12 1695 . . . . . . . . . 10  |-  ( z  =  u  ->  ( [ w  /  y ] ps  <->  [ u  /  z ] [ w  /  y ] ps ) )
298, 28sylan9bb 450 . . . . . . . . 9  |-  ( ( y  =  w  /\  z  =  u )  ->  ( ps  <->  [ u  /  z ] [
w  /  y ] ps ) )
3027, 29sylbi 119 . . . . . . . 8  |-  ( <.
y ,  z >.  =  <. w ,  u >.  ->  ( ps  <->  [ u  /  z ] [
w  /  y ] ps ) )
3124, 30sylan9bb 450 . . . . . . 7  |-  ( ( v  =  <. y ,  z >.  /\  <. y ,  z >.  =  <. w ,  u >. )  ->  ( [ v  /  x ] ph  <->  [ u  /  z ] [
w  /  y ] ps ) )
3221, 31exlimi 1526 . . . . . 6  |-  ( E. z ( v  = 
<. y ,  z >.  /\  <. y ,  z
>.  =  <. w ,  u >. )  ->  ( [ v  /  x ] ph  <->  [ u  /  z ] [ w  /  y ] ps ) )
3317, 32exlimi 1526 . . . . 5  |-  ( E. y E. z ( v  =  <. y ,  z >.  /\  <. y ,  z >.  =  <. w ,  u >. )  ->  ( [ v  /  x ] ph  <->  [ u  /  z ] [
w  /  y ] ps ) )
3413, 33sylbi 119 . . . 4  |-  ( v  =  <. w ,  u >.  ->  ( [ v  /  x ] ph  <->  [ u  /  z ] [ w  /  y ] ps ) )
3534rexxp 4508 . . 3  |-  ( E. v  e.  ( A  X.  B ) [ v  /  x ] ph 
<->  E. w  e.  A  E. u  e.  B  [ u  /  z ] [ w  /  y ] ps )
363, 10, 353bitr4ri 211 . 2  |-  ( E. v  e.  ( A  X.  B ) [ v  /  x ] ph 
<->  E. y  e.  A  E. z  e.  B  ps )
371, 36bitri 182 1  |-  ( E. x  e.  ( A  X.  B ) ph  <->  E. y  e.  A  E. z  e.  B  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1285   F/wnf 1390   E.wex 1422   [wsb 1686   E.wrex 2350   <.cop 3409    X. cxp 4369
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-sbc 2817  df-csb 2910  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-iun 3688  df-opab 3848  df-xp 4377  df-rel 4378
This theorem is referenced by:  iunxpf  4512
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