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Theorem ralxpf 5154
Description: Version of ralxp 5149 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 3104 . 2  |-  ( A. x  e.  ( A  X.  B ) ph  <->  A. w  e.  ( A  X.  B
) [ w  /  x ] ph )
2 cbvralsv 3104 . . . 4  |-  ( A. z  e.  B  [
u  /  y ] ps  <->  A. v  e.  B  [ v  /  z ] [ u  /  y ] ps )
32ralbii 2898 . . 3  |-  ( A. u  e.  A  A. z  e.  B  [
u  /  y ] ps  <->  A. u  e.  A  A. v  e.  B  [ v  /  z ] [ u  /  y ] ps )
4 nfv 1683 . . . 4  |-  F/ u A. z  e.  B  ps
5 nfcv 2629 . . . . 5  |-  F/_ y B
6 nfs1v 2164 . . . . 5  |-  F/ y [ u  /  y ] ps
75, 6nfral 2853 . . . 4  |-  F/ y A. z  e.  B  [ u  /  y ] ps
8 sbequ12 1961 . . . . 5  |-  ( y  =  u  ->  ( ps 
<->  [ u  /  y ] ps ) )
98ralbidv 2906 . . . 4  |-  ( y  =  u  ->  ( A. z  e.  B  ps 
<-> 
A. z  e.  B  [ u  /  y ] ps ) )
104, 7, 9cbvral 3089 . . 3  |-  ( A. y  e.  A  A. z  e.  B  ps  <->  A. u  e.  A  A. z  e.  B  [
u  /  y ] ps )
11 vex 3121 . . . . . 6  |-  u  e. 
_V
12 vex 3121 . . . . . 6  |-  v  e. 
_V
1311, 12eqvinop 4736 . . . . 5  |-  ( w  =  <. u ,  v
>. 
<->  E. y E. z
( w  =  <. y ,  z >.  /\  <. y ,  z >.  =  <. u ,  v >. )
)
14 ralxpf.1 . . . . . . . 8  |-  F/ y
ph
1514nfsb 2168 . . . . . . 7  |-  F/ y [ w  /  x ] ph
166nfsb 2168 . . . . . . 7  |-  F/ y [ v  /  z ] [ u  /  y ] ps
1715, 16nfbi 1881 . . . . . 6  |-  F/ y ( [ w  /  x ] ph  <->  [ v  /  z ] [
u  /  y ] ps )
18 ralxpf.2 . . . . . . . . 9  |-  F/ z
ph
1918nfsb 2168 . . . . . . . 8  |-  F/ z [ w  /  x ] ph
20 nfs1v 2164 . . . . . . . 8  |-  F/ z [ v  /  z ] [ u  /  y ] ps
2119, 20nfbi 1881 . . . . . . 7  |-  F/ z ( [ w  /  x ] ph  <->  [ v  /  z ] [
u  /  y ] ps )
22 ralxpf.3 . . . . . . . . 9  |-  F/ x ps
23 ralxpf.4 . . . . . . . . 9  |-  ( x  =  <. y ,  z
>.  ->  ( ph  <->  ps )
)
2422, 23sbhypf 3165 . . . . . . . 8  |-  ( w  =  <. y ,  z
>.  ->  ( [ w  /  x ] ph  <->  ps )
)
25 vex 3121 . . . . . . . . . 10  |-  y  e. 
_V
26 vex 3121 . . . . . . . . . 10  |-  z  e. 
_V
2725, 26opth 4726 . . . . . . . . 9  |-  ( <.
y ,  z >.  =  <. u ,  v
>. 
<->  ( y  =  u  /\  z  =  v ) )
28 sbequ12 1961 . . . . . . . . . 10  |-  ( z  =  v  ->  ( [ u  /  y ] ps  <->  [ v  /  z ] [ u  /  y ] ps ) )
298, 28sylan9bb 699 . . . . . . . . 9  |-  ( ( y  =  u  /\  z  =  v )  ->  ( ps  <->  [ v  /  z ] [
u  /  y ] ps ) )
3027, 29sylbi 195 . . . . . . . 8  |-  ( <.
y ,  z >.  =  <. u ,  v
>.  ->  ( ps  <->  [ v  /  z ] [
u  /  y ] ps ) )
3124, 30sylan9bb 699 . . . . . . 7  |-  ( ( w  =  <. y ,  z >.  /\  <. y ,  z >.  =  <. u ,  v >. )  ->  ( [ w  /  x ] ph  <->  [ v  /  z ] [
u  /  y ] ps ) )
3221, 31exlimi 1859 . . . . . 6  |-  ( E. z ( w  = 
<. y ,  z >.  /\  <. y ,  z
>.  =  <. u ,  v >. )  ->  ( [ w  /  x ] ph  <->  [ v  /  z ] [ u  /  y ] ps ) )
3317, 32exlimi 1859 . . . . 5  |-  ( E. y E. z ( w  =  <. y ,  z >.  /\  <. y ,  z >.  =  <. u ,  v >. )  ->  ( [ w  /  x ] ph  <->  [ v  /  z ] [
u  /  y ] ps ) )
3413, 33sylbi 195 . . . 4  |-  ( w  =  <. u ,  v
>.  ->  ( [ w  /  x ] ph  <->  [ v  /  z ] [
u  /  y ] ps ) )
3534ralxp 5149 . . 3  |-  ( A. w  e.  ( A  X.  B ) [ w  /  x ] ph  <->  A. u  e.  A  A. v  e.  B  [ v  /  z ] [
u  /  y ] ps )
363, 10, 353bitr4ri 278 . 2  |-  ( A. w  e.  ( A  X.  B ) [ w  /  x ] ph  <->  A. y  e.  A  A. z  e.  B  ps )
371, 36bitri 249 1  |-  ( A. x  e.  ( A  X.  B ) ph  <->  A. y  e.  A  A. z  e.  B  ps )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379   E.wex 1596   F/wnf 1599   [wsb 1711   A.wral 2817   <.cop 4038    X. cxp 5002
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-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4573  ax-nul 4581  ax-pr 4691
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-iun 4332  df-opab 4511  df-xp 5010  df-rel 5011
This theorem is referenced by:  rexxpf  5155
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