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

Theorem cbvralcsf 3275
Description: A more general version of cbvralf 2890 that doesn't require  A and  B to be distinct from  x or  y. Changes bound variables using implicit substitution. (Contributed by Andrew Salmon, 13-Jul-2011.)
Hypotheses
Ref Expression
cbvralcsf.1  |-  F/_ y A
cbvralcsf.2  |-  F/_ x B
cbvralcsf.3  |-  F/ y
ph
cbvralcsf.4  |-  F/ x ps
cbvralcsf.5  |-  ( x  =  y  ->  A  =  B )
cbvralcsf.6  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
cbvralcsf  |-  ( A. x  e.  A  ph  <->  A. y  e.  B  ps )

Proof of Theorem cbvralcsf
Dummy variables  v 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfv 1626 . . . 4  |-  F/ z ( x  e.  A  ->  ph )
2 nfcsb1v 3247 . . . . . 6  |-  F/_ x [_ z  /  x ]_ A
32nfcri 2538 . . . . 5  |-  F/ x  z  e.  [_ z  /  x ]_ A
4 nfsbc1v 3144 . . . . 5  |-  F/ x [. z  /  x ]. ph
53, 4nfim 1828 . . . 4  |-  F/ x
( z  e.  [_ z  /  x ]_ A  ->  [. z  /  x ]. ph )
6 id 20 . . . . . 6  |-  ( x  =  z  ->  x  =  z )
7 csbeq1a 3223 . . . . . 6  |-  ( x  =  z  ->  A  =  [_ z  /  x ]_ A )
86, 7eleq12d 2476 . . . . 5  |-  ( x  =  z  ->  (
x  e.  A  <->  z  e.  [_ z  /  x ]_ A ) )
9 sbceq1a 3135 . . . . 5  |-  ( x  =  z  ->  ( ph 
<-> 
[. z  /  x ]. ph ) )
108, 9imbi12d 312 . . . 4  |-  ( x  =  z  ->  (
( x  e.  A  ->  ph )  <->  ( z  e.  [_ z  /  x ]_ A  ->  [. z  /  x ]. ph )
) )
111, 5, 10cbval 2040 . . 3  |-  ( A. x ( x  e.  A  ->  ph )  <->  A. z
( z  e.  [_ z  /  x ]_ A  ->  [. z  /  x ]. ph ) )
12 nfcv 2544 . . . . . . 7  |-  F/_ y
z
13 cbvralcsf.1 . . . . . . 7  |-  F/_ y A
1412, 13nfcsb 3249 . . . . . 6  |-  F/_ y [_ z  /  x ]_ A
1514nfcri 2538 . . . . 5  |-  F/ y  z  e.  [_ z  /  x ]_ A
16 cbvralcsf.3 . . . . . 6  |-  F/ y
ph
1712, 16nfsbc 3146 . . . . 5  |-  F/ y
[. z  /  x ]. ph
1815, 17nfim 1828 . . . 4  |-  F/ y ( z  e.  [_ z  /  x ]_ A  ->  [. z  /  x ]. ph )
19 nfv 1626 . . . 4  |-  F/ z ( y  e.  B  ->  ps )
20 id 20 . . . . . 6  |-  ( z  =  y  ->  z  =  y )
21 csbeq1 3218 . . . . . . 7  |-  ( z  =  y  ->  [_ z  /  x ]_ A  = 
[_ y  /  x ]_ A )
22 df-csb 3216 . . . . . . . 8  |-  [_ y  /  x ]_ A  =  { v  |  [. y  /  x ]. v  e.  A }
23 cbvralcsf.2 . . . . . . . . . . . 12  |-  F/_ x B
2423nfcri 2538 . . . . . . . . . . 11  |-  F/ x  v  e.  B
25 cbvralcsf.5 . . . . . . . . . . . 12  |-  ( x  =  y  ->  A  =  B )
2625eleq2d 2475 . . . . . . . . . . 11  |-  ( x  =  y  ->  (
v  e.  A  <->  v  e.  B ) )
2724, 26sbie 2091 . . . . . . . . . 10  |-  ( [ y  /  x ]
v  e.  A  <->  v  e.  B )
28 sbsbc 3129 . . . . . . . . . 10  |-  ( [ y  /  x ]
v  e.  A  <->  [. y  /  x ]. v  e.  A
)
2927, 28bitr3i 243 . . . . . . . . 9  |-  ( v  e.  B  <->  [. y  /  x ]. v  e.  A
)
3029abbi2i 2519 . . . . . . . 8  |-  B  =  { v  |  [. y  /  x ]. v  e.  A }
3122, 30eqtr4i 2431 . . . . . . 7  |-  [_ y  /  x ]_ A  =  B
3221, 31syl6eq 2456 . . . . . 6  |-  ( z  =  y  ->  [_ z  /  x ]_ A  =  B )
3320, 32eleq12d 2476 . . . . 5  |-  ( z  =  y  ->  (
z  e.  [_ z  /  x ]_ A  <->  y  e.  B ) )
34 dfsbcq 3127 . . . . . 6  |-  ( z  =  y  ->  ( [. z  /  x ]. ph  <->  [. y  /  x ]. ph ) )
35 sbsbc 3129 . . . . . . 7  |-  ( [ y  /  x ] ph 
<-> 
[. y  /  x ]. ph )
36 cbvralcsf.4 . . . . . . . 8  |-  F/ x ps
37 cbvralcsf.6 . . . . . . . 8  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
3836, 37sbie 2091 . . . . . . 7  |-  ( [ y  /  x ] ph 
<->  ps )
3935, 38bitr3i 243 . . . . . 6  |-  ( [. y  /  x ]. ph  <->  ps )
4034, 39syl6bb 253 . . . . 5  |-  ( z  =  y  ->  ( [. z  /  x ]. ph  <->  ps ) )
4133, 40imbi12d 312 . . . 4  |-  ( z  =  y  ->  (
( z  e.  [_ z  /  x ]_ A  ->  [. z  /  x ]. ph )  <->  ( y  e.  B  ->  ps )
) )
4218, 19, 41cbval 2040 . . 3  |-  ( A. z ( z  e. 
[_ z  /  x ]_ A  ->  [. z  /  x ]. ph )  <->  A. y ( y  e.  B  ->  ps )
)
4311, 42bitri 241 . 2  |-  ( A. x ( x  e.  A  ->  ph )  <->  A. y
( y  e.  B  ->  ps ) )
44 df-ral 2675 . 2  |-  ( A. x  e.  A  ph  <->  A. x
( x  e.  A  ->  ph ) )
45 df-ral 2675 . 2  |-  ( A. y  e.  B  ps  <->  A. y ( y  e.  B  ->  ps )
)
4643, 44, 453bitr4i 269 1  |-  ( A. x  e.  A  ph  <->  A. y  e.  B  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177   A.wal 1546   F/wnf 1550    = wceq 1649   [wsb 1655    e. wcel 1721   {cab 2394   F/_wnfc 2531   A.wral 2670   [.wsbc 3125   [_csb 3215
This theorem is referenced by:  cbvrexcsf  3276  cbvralv2  3279
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ral 2675  df-sbc 3126  df-csb 3216
  Copyright terms: Public domain W3C validator