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Theorem ralxpf 4981
Description: Version of ralxp 4976 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 3030 . 2  |-  ( A. x  e.  ( A  X.  B ) ph  <->  A. w  e.  ( A  X.  B
) [ w  /  x ] ph )
2 cbvralsv 3030 . . . 4  |-  ( A. z  e.  B  [
u  /  y ] ps  <->  A. v  e.  B  [ v  /  z ] [ u  /  y ] ps )
32ralbii 2819 . . 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 1761 . . . 4  |-  F/ u A. z  e.  B  ps
5 nfcv 2592 . . . . 5  |-  F/_ y B
6 nfs1v 2266 . . . . 5  |-  F/ y [ u  /  y ] ps
75, 6nfral 2774 . . . 4  |-  F/ y A. z  e.  B  [ u  /  y ] ps
8 sbequ12 2083 . . . . 5  |-  ( y  =  u  ->  ( ps 
<->  [ u  /  y ] ps ) )
98ralbidv 2827 . . . 4  |-  ( y  =  u  ->  ( A. z  e.  B  ps 
<-> 
A. z  e.  B  [ u  /  y ] ps ) )
104, 7, 9cbvral 3015 . . 3  |-  ( A. y  e.  A  A. z  e.  B  ps  <->  A. u  e.  A  A. z  e.  B  [
u  /  y ] ps )
11 vex 3048 . . . . . 6  |-  u  e. 
_V
12 vex 3048 . . . . . 6  |-  v  e. 
_V
1311, 12eqvinop 4686 . . . . 5  |-  ( w  =  <. u ,  v
>. 
<->  E. y E. z
( w  =  <. y ,  z >.  /\  <. y ,  z >.  =  <. u ,  v >. )
)
14 ralxpf.1 . . . . . . . 8  |-  F/ y
ph
1514nfsb 2269 . . . . . . 7  |-  F/ y [ w  /  x ] ph
166nfsb 2269 . . . . . . 7  |-  F/ y [ v  /  z ] [ u  /  y ] ps
1715, 16nfbi 2017 . . . . . 6  |-  F/ y ( [ w  /  x ] ph  <->  [ v  /  z ] [
u  /  y ] ps )
18 ralxpf.2 . . . . . . . . 9  |-  F/ z
ph
1918nfsb 2269 . . . . . . . 8  |-  F/ z [ w  /  x ] ph
20 nfs1v 2266 . . . . . . . 8  |-  F/ z [ v  /  z ] [ u  /  y ] ps
2119, 20nfbi 2017 . . . . . . 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 3095 . . . . . . . 8  |-  ( w  =  <. y ,  z
>.  ->  ( [ w  /  x ] ph  <->  ps )
)
25 vex 3048 . . . . . . . . . 10  |-  y  e. 
_V
26 vex 3048 . . . . . . . . . 10  |-  z  e. 
_V
2725, 26opth 4676 . . . . . . . . 9  |-  ( <.
y ,  z >.  =  <. u ,  v
>. 
<->  ( y  =  u  /\  z  =  v ) )
28 sbequ12 2083 . . . . . . . . . 10  |-  ( z  =  v  ->  ( [ u  /  y ] ps  <->  [ v  /  z ] [ u  /  y ] ps ) )
298, 28sylan9bb 706 . . . . . . . . 9  |-  ( ( y  =  u  /\  z  =  v )  ->  ( ps  <->  [ v  /  z ] [
u  /  y ] ps ) )
3027, 29sylbi 199 . . . . . . . 8  |-  ( <.
y ,  z >.  =  <. u ,  v
>.  ->  ( ps  <->  [ v  /  z ] [
u  /  y ] ps ) )
3124, 30sylan9bb 706 . . . . . . 7  |-  ( ( w  =  <. y ,  z >.  /\  <. y ,  z >.  =  <. u ,  v >. )  ->  ( [ w  /  x ] ph  <->  [ v  /  z ] [
u  /  y ] ps ) )
3221, 31exlimi 1995 . . . . . 6  |-  ( E. z ( w  = 
<. y ,  z >.  /\  <. y ,  z
>.  =  <. u ,  v >. )  ->  ( [ w  /  x ] ph  <->  [ v  /  z ] [ u  /  y ] ps ) )
3317, 32exlimi 1995 . . . . 5  |-  ( E. y E. z ( w  =  <. y ,  z >.  /\  <. y ,  z >.  =  <. u ,  v >. )  ->  ( [ w  /  x ] ph  <->  [ v  /  z ] [
u  /  y ] ps ) )
3413, 33sylbi 199 . . . 4  |-  ( w  =  <. u ,  v
>.  ->  ( [ w  /  x ] ph  <->  [ v  /  z ] [
u  /  y ] ps ) )
3534ralxp 4976 . . 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 282 . 2  |-  ( A. w  e.  ( A  X.  B ) [ w  /  x ] ph  <->  A. y  e.  A  A. z  e.  B  ps )
371, 36bitri 253 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 188    /\ wa 371    = wceq 1444   E.wex 1663   F/wnf 1667   [wsb 1797   A.wral 2737   <.cop 3974    X. cxp 4832
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pr 4639
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-iun 4280  df-opab 4462  df-xp 4840  df-rel 4841
This theorem is referenced by:  rexxpf  4982
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