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Theorem ralxpf 5019
Description: Version of ralxp 5016 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 2943 . 2  |-  ( A. x  e.  ( A  X.  B ) ph  <->  A. w  e.  ( A  X.  B
) [ w  /  x ] ph )
2 cbvralsv 2943 . . . 4  |-  ( A. z  e.  B  [
u  /  y ] ps  <->  A. v  e.  B  [ v  /  z ] [ u  /  y ] ps )
32ralbii 2729 . . 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 1629 . . . 4  |-  F/ u A. z  e.  B  ps
5 nfcv 2572 . . . . 5  |-  F/_ y B
6 nfs1v 2182 . . . . 5  |-  F/ y [ u  /  y ] ps
75, 6nfral 2759 . . . 4  |-  F/ y A. z  e.  B  [ u  /  y ] ps
8 sbequ12 1944 . . . . 5  |-  ( y  =  u  ->  ( ps 
<->  [ u  /  y ] ps ) )
98ralbidv 2725 . . . 4  |-  ( y  =  u  ->  ( A. z  e.  B  ps 
<-> 
A. z  e.  B  [ u  /  y ] ps ) )
104, 7, 9cbvral 2928 . . 3  |-  ( A. y  e.  A  A. z  e.  B  ps  <->  A. u  e.  A  A. z  e.  B  [
u  /  y ] ps )
11 vex 2959 . . . . . 6  |-  u  e. 
_V
12 vex 2959 . . . . . 6  |-  v  e. 
_V
1311, 12eqvinop 4441 . . . . 5  |-  ( w  =  <. u ,  v
>. 
<->  E. y E. z
( w  =  <. y ,  z >.  /\  <. y ,  z >.  =  <. u ,  v >. )
)
14 ralxpf.1 . . . . . . . 8  |-  F/ y
ph
1514nfsb 2185 . . . . . . 7  |-  F/ y [ w  /  x ] ph
166nfsb 2185 . . . . . . 7  |-  F/ y [ v  /  z ] [ u  /  y ] ps
1715, 16nfbi 1856 . . . . . 6  |-  F/ y ( [ w  /  x ] ph  <->  [ v  /  z ] [
u  /  y ] ps )
18 ralxpf.2 . . . . . . . . 9  |-  F/ z
ph
1918nfsb 2185 . . . . . . . 8  |-  F/ z [ w  /  x ] ph
20 nfs1v 2182 . . . . . . . 8  |-  F/ z [ v  /  z ] [ u  /  y ] ps
2119, 20nfbi 1856 . . . . . . 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 3001 . . . . . . . 8  |-  ( w  =  <. y ,  z
>.  ->  ( [ w  /  x ] ph  <->  ps )
)
25 vex 2959 . . . . . . . . . 10  |-  y  e. 
_V
26 vex 2959 . . . . . . . . . 10  |-  z  e. 
_V
2725, 26opth 4435 . . . . . . . . 9  |-  ( <.
y ,  z >.  =  <. u ,  v
>. 
<->  ( y  =  u  /\  z  =  v ) )
28 sbequ12 1944 . . . . . . . . . 10  |-  ( z  =  v  ->  ( [ u  /  y ] ps  <->  [ v  /  z ] [ u  /  y ] ps ) )
298, 28sylan9bb 681 . . . . . . . . 9  |-  ( ( y  =  u  /\  z  =  v )  ->  ( ps  <->  [ v  /  z ] [
u  /  y ] ps ) )
3027, 29sylbi 188 . . . . . . . 8  |-  ( <.
y ,  z >.  =  <. u ,  v
>.  ->  ( ps  <->  [ v  /  z ] [
u  /  y ] ps ) )
3124, 30sylan9bb 681 . . . . . . 7  |-  ( ( w  =  <. y ,  z >.  /\  <. y ,  z >.  =  <. u ,  v >. )  ->  ( [ w  /  x ] ph  <->  [ v  /  z ] [
u  /  y ] ps ) )
3221, 31exlimi 1821 . . . . . 6  |-  ( E. z ( w  = 
<. y ,  z >.  /\  <. y ,  z
>.  =  <. u ,  v >. )  ->  ( [ w  /  x ] ph  <->  [ v  /  z ] [ u  /  y ] ps ) )
3317, 32exlimi 1821 . . . . 5  |-  ( E. y E. z ( w  =  <. y ,  z >.  /\  <. y ,  z >.  =  <. u ,  v >. )  ->  ( [ w  /  x ] ph  <->  [ v  /  z ] [
u  /  y ] ps ) )
3413, 33sylbi 188 . . . 4  |-  ( w  =  <. u ,  v
>.  ->  ( [ w  /  x ] ph  <->  [ v  /  z ] [
u  /  y ] ps ) )
3534ralxp 5016 . . 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 270 . 2  |-  ( A. w  e.  ( A  X.  B ) [ w  /  x ] ph  <->  A. y  e.  A  A. z  e.  B  ps )
371, 36bitri 241 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    <-> wb 177    /\ wa 359   E.wex 1550   F/wnf 1553    = wceq 1652   [wsb 1658   A.wral 2705   <.cop 3817    X. cxp 4876
This theorem is referenced by:  rexxpf  5020
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-iun 4095  df-opab 4267  df-xp 4884  df-rel 4885
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