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Theorem ralxpf 4510
Description: Version of ralxp 4507 with bound-variable hypotheses. (Contributed by NM, 18-Aug-2006.) (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
ralxpf  |-  ( A. x  e.  ( A  X.  B ) ph  <->  A. y  e.  A  A. 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 ralxpf
Dummy variables  v  u  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cbvralsv 2561 . 2  |-  ( A. x  e.  ( A  X.  B ) ph  <->  A. v  e.  ( A  X.  B
) [ v  /  x ] ph )
2 cbvralsv 2561 . . . 4  |-  ( A. z  e.  B  [
w  /  y ] ps  <->  A. u  e.  B  [ u  /  z ] [ w  /  y ] ps )
32ralbii 2347 . . 3  |-  ( A. w  e.  A  A. z  e.  B  [
w  /  y ] ps  <->  A. w  e.  A  A. u  e.  B  [ u  /  z ] [ w  /  y ] ps )
4 nfv 1437 . . . 4  |-  F/ w A. z  e.  B  ps
5 nfcv 2194 . . . . 5  |-  F/_ y B
6 nfs1v 1831 . . . . 5  |-  F/ y [ w  /  y ] ps
75, 6nfralxy 2377 . . . 4  |-  F/ y A. z  e.  B  [ w  /  y ] ps
8 sbequ12 1670 . . . . 5  |-  ( y  =  w  ->  ( ps 
<->  [ w  /  y ] ps ) )
98ralbidv 2343 . . . 4  |-  ( y  =  w  ->  ( A. z  e.  B  ps 
<-> 
A. z  e.  B  [ w  /  y ] ps ) )
104, 7, 9cbvral 2546 . . 3  |-  ( A. y  e.  A  A. z  e.  B  ps  <->  A. w  e.  A  A. z  e.  B  [
w  /  y ] ps )
11 vex 2577 . . . . . 6  |-  w  e. 
_V
12 vex 2577 . . . . . 6  |-  u  e. 
_V
1311, 12eqvinop 4008 . . . . 5  |-  ( v  =  <. w ,  u >.  <->  E. y E. z ( v  =  <. y ,  z >.  /\  <. y ,  z >.  =  <. w ,  u >. )
)
14 ralxpf.1 . . . . . . . 8  |-  F/ y
ph
1514nfsb 1838 . . . . . . 7  |-  F/ y [ v  /  x ] ph
166nfsb 1838 . . . . . . 7  |-  F/ y [ u  /  z ] [ w  /  y ] ps
1715, 16nfbi 1497 . . . . . 6  |-  F/ y ( [ v  /  x ] ph  <->  [ u  /  z ] [
w  /  y ] ps )
18 ralxpf.2 . . . . . . . . 9  |-  F/ z
ph
1918nfsb 1838 . . . . . . . 8  |-  F/ z [ v  /  x ] ph
20 nfs1v 1831 . . . . . . . 8  |-  F/ z [ u  /  z ] [ w  /  y ] ps
2119, 20nfbi 1497 . . . . . . 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 2620 . . . . . . . 8  |-  ( v  =  <. y ,  z
>.  ->  ( [ v  /  x ] ph  <->  ps ) )
25 vex 2577 . . . . . . . . . 10  |-  y  e. 
_V
26 vex 2577 . . . . . . . . . 10  |-  z  e. 
_V
2725, 26opth 4002 . . . . . . . . 9  |-  ( <.
y ,  z >.  =  <. w ,  u >.  <-> 
( y  =  w  /\  z  =  u ) )
28 sbequ12 1670 . . . . . . . . . 10  |-  ( z  =  u  ->  ( [ w  /  y ] ps  <->  [ u  /  z ] [ w  /  y ] ps ) )
298, 28sylan9bb 443 . . . . . . . . 9  |-  ( ( y  =  w  /\  z  =  u )  ->  ( ps  <->  [ u  /  z ] [
w  /  y ] ps ) )
3027, 29sylbi 118 . . . . . . . 8  |-  ( <.
y ,  z >.  =  <. w ,  u >.  ->  ( ps  <->  [ u  /  z ] [
w  /  y ] ps ) )
3124, 30sylan9bb 443 . . . . . . 7  |-  ( ( v  =  <. y ,  z >.  /\  <. y ,  z >.  =  <. w ,  u >. )  ->  ( [ v  /  x ] ph  <->  [ u  /  z ] [
w  /  y ] ps ) )
3221, 31exlimi 1501 . . . . . 6  |-  ( E. z ( v  = 
<. y ,  z >.  /\  <. y ,  z
>.  =  <. w ,  u >. )  ->  ( [ v  /  x ] ph  <->  [ u  /  z ] [ w  /  y ] ps ) )
3317, 32exlimi 1501 . . . . 5  |-  ( E. y E. z ( v  =  <. y ,  z >.  /\  <. y ,  z >.  =  <. w ,  u >. )  ->  ( [ v  /  x ] ph  <->  [ u  /  z ] [
w  /  y ] ps ) )
3413, 33sylbi 118 . . . 4  |-  ( v  =  <. w ,  u >.  ->  ( [ v  /  x ] ph  <->  [ u  /  z ] [ w  /  y ] ps ) )
3534ralxp 4507 . . 3  |-  ( A. v  e.  ( A  X.  B ) [ v  /  x ] ph  <->  A. w  e.  A  A. u  e.  B  [
u  /  z ] [ w  /  y ] ps )
363, 10, 353bitr4ri 206 . 2  |-  ( A. v  e.  ( A  X.  B ) [ v  /  x ] ph  <->  A. y  e.  A  A. z  e.  B  ps )
371, 36bitri 177 1  |-  ( A. x  e.  ( A  X.  B ) ph  <->  A. y  e.  A  A. z  e.  B  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    <-> wb 102    = wceq 1259   F/wnf 1365   E.wex 1397   [wsb 1661   A.wral 2323   <.cop 3406    X. cxp 4371
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-sbc 2788  df-csb 2881  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-iun 3687  df-opab 3847  df-xp 4379  df-rel 4380
This theorem is referenced by: (None)
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