MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ralxpf Structured version   Unicode version

Theorem ralxpf 5138
Description: Version of ralxp 5133 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 3092 . 2  |-  ( A. x  e.  ( A  X.  B ) ph  <->  A. w  e.  ( A  X.  B
) [ w  /  x ] ph )
2 cbvralsv 3092 . . . 4  |-  ( A. z  e.  B  [
u  /  y ] ps  <->  A. v  e.  B  [ v  /  z ] [ u  /  y ] ps )
32ralbii 2885 . . 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 1712 . . . 4  |-  F/ u A. z  e.  B  ps
5 nfcv 2616 . . . . 5  |-  F/_ y B
6 nfs1v 2183 . . . . 5  |-  F/ y [ u  /  y ] ps
75, 6nfral 2840 . . . 4  |-  F/ y A. z  e.  B  [ u  /  y ] ps
8 sbequ12 1997 . . . . 5  |-  ( y  =  u  ->  ( ps 
<->  [ u  /  y ] ps ) )
98ralbidv 2893 . . . 4  |-  ( y  =  u  ->  ( A. z  e.  B  ps 
<-> 
A. z  e.  B  [ u  /  y ] ps ) )
104, 7, 9cbvral 3077 . . 3  |-  ( A. y  e.  A  A. z  e.  B  ps  <->  A. u  e.  A  A. z  e.  B  [
u  /  y ] ps )
11 vex 3109 . . . . . 6  |-  u  e. 
_V
12 vex 3109 . . . . . 6  |-  v  e. 
_V
1311, 12eqvinop 4721 . . . . 5  |-  ( w  =  <. u ,  v
>. 
<->  E. y E. z
( w  =  <. y ,  z >.  /\  <. y ,  z >.  =  <. u ,  v >. )
)
14 ralxpf.1 . . . . . . . 8  |-  F/ y
ph
1514nfsb 2186 . . . . . . 7  |-  F/ y [ w  /  x ] ph
166nfsb 2186 . . . . . . 7  |-  F/ y [ v  /  z ] [ u  /  y ] ps
1715, 16nfbi 1939 . . . . . 6  |-  F/ y ( [ w  /  x ] ph  <->  [ v  /  z ] [
u  /  y ] ps )
18 ralxpf.2 . . . . . . . . 9  |-  F/ z
ph
1918nfsb 2186 . . . . . . . 8  |-  F/ z [ w  /  x ] ph
20 nfs1v 2183 . . . . . . . 8  |-  F/ z [ v  /  z ] [ u  /  y ] ps
2119, 20nfbi 1939 . . . . . . 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 3153 . . . . . . . 8  |-  ( w  =  <. y ,  z
>.  ->  ( [ w  /  x ] ph  <->  ps )
)
25 vex 3109 . . . . . . . . . 10  |-  y  e. 
_V
26 vex 3109 . . . . . . . . . 10  |-  z  e. 
_V
2725, 26opth 4711 . . . . . . . . 9  |-  ( <.
y ,  z >.  =  <. u ,  v
>. 
<->  ( y  =  u  /\  z  =  v ) )
28 sbequ12 1997 . . . . . . . . . 10  |-  ( z  =  v  ->  ( [ u  /  y ] ps  <->  [ v  /  z ] [ u  /  y ] ps ) )
298, 28sylan9bb 697 . . . . . . . . 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 697 . . . . . . 7  |-  ( ( w  =  <. y ,  z >.  /\  <. y ,  z >.  =  <. u ,  v >. )  ->  ( [ w  /  x ] ph  <->  [ v  /  z ] [
u  /  y ] ps ) )
3221, 31exlimi 1917 . . . . . 6  |-  ( E. z ( w  = 
<. y ,  z >.  /\  <. y ,  z
>.  =  <. u ,  v >. )  ->  ( [ w  /  x ] ph  <->  [ v  /  z ] [ u  /  y ] ps ) )
3317, 32exlimi 1917 . . . . 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 5133 . . 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 367    = wceq 1398   E.wex 1617   F/wnf 1621   [wsb 1744   A.wral 2804   <.cop 4022    X. cxp 4986
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-iun 4317  df-opab 4498  df-xp 4994  df-rel 4995
This theorem is referenced by:  rexxpf  5139
  Copyright terms: Public domain W3C validator