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

Theorem cbvralcsf 3143
Description: A more general version of cbvralf 2758 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 1605 . . . 4  |-  F/ z ( x  e.  A  ->  ph )
2 nfcsb1v 3113 . . . . . 6  |-  F/_ x [_ z  /  x ]_ A
32nfcri 2413 . . . . 5  |-  F/ x  z  e.  [_ z  /  x ]_ A
4 nfsbc1v 3010 . . . . 5  |-  F/ x [. z  /  x ]. ph
53, 4nfim 1769 . . . 4  |-  F/ x
( z  e.  [_ z  /  x ]_ A  ->  [. z  /  x ]. ph )
6 id 19 . . . . . 6  |-  ( x  =  z  ->  x  =  z )
7 csbeq1a 3089 . . . . . 6  |-  ( x  =  z  ->  A  =  [_ z  /  x ]_ A )
86, 7eleq12d 2351 . . . . 5  |-  ( x  =  z  ->  (
x  e.  A  <->  z  e.  [_ z  /  x ]_ A ) )
9 sbceq1a 3001 . . . . 5  |-  ( x  =  z  ->  ( ph 
<-> 
[. z  /  x ]. ph ) )
108, 9imbi12d 311 . . . 4  |-  ( x  =  z  ->  (
( x  e.  A  ->  ph )  <->  ( z  e.  [_ z  /  x ]_ A  ->  [. z  /  x ]. ph )
) )
111, 5, 10cbval 1924 . . 3  |-  ( A. x ( x  e.  A  ->  ph )  <->  A. z
( z  e.  [_ z  /  x ]_ A  ->  [. z  /  x ]. ph ) )
12 nfcv 2419 . . . . . . 7  |-  F/_ y
z
13 cbvralcsf.1 . . . . . . 7  |-  F/_ y A
1412, 13nfcsb 3115 . . . . . 6  |-  F/_ y [_ z  /  x ]_ A
1514nfcri 2413 . . . . 5  |-  F/ y  z  e.  [_ z  /  x ]_ A
16 cbvralcsf.3 . . . . . 6  |-  F/ y
ph
1712, 16nfsbc 3012 . . . . 5  |-  F/ y
[. z  /  x ]. ph
1815, 17nfim 1769 . . . 4  |-  F/ y ( z  e.  [_ z  /  x ]_ A  ->  [. z  /  x ]. ph )
19 nfv 1605 . . . 4  |-  F/ z ( y  e.  B  ->  ps )
20 id 19 . . . . . 6  |-  ( z  =  y  ->  z  =  y )
21 csbeq1 3084 . . . . . . 7  |-  ( z  =  y  ->  [_ z  /  x ]_ A  = 
[_ y  /  x ]_ A )
22 df-csb 3082 . . . . . . . 8  |-  [_ y  /  x ]_ A  =  { v  |  [. y  /  x ]. v  e.  A }
23 cbvralcsf.2 . . . . . . . . . . . 12  |-  F/_ x B
2423nfcri 2413 . . . . . . . . . . 11  |-  F/ x  v  e.  B
25 cbvralcsf.5 . . . . . . . . . . . 12  |-  ( x  =  y  ->  A  =  B )
2625eleq2d 2350 . . . . . . . . . . 11  |-  ( x  =  y  ->  (
v  e.  A  <->  v  e.  B ) )
2724, 26sbie 1978 . . . . . . . . . 10  |-  ( [ y  /  x ]
v  e.  A  <->  v  e.  B )
28 sbsbc 2995 . . . . . . . . . 10  |-  ( [ y  /  x ]
v  e.  A  <->  [. y  /  x ]. v  e.  A
)
2927, 28bitr3i 242 . . . . . . . . 9  |-  ( v  e.  B  <->  [. y  /  x ]. v  e.  A
)
3029abbi2i 2394 . . . . . . . 8  |-  B  =  { v  |  [. y  /  x ]. v  e.  A }
3122, 30eqtr4i 2306 . . . . . . 7  |-  [_ y  /  x ]_ A  =  B
3221, 31syl6eq 2331 . . . . . 6  |-  ( z  =  y  ->  [_ z  /  x ]_ A  =  B )
3320, 32eleq12d 2351 . . . . 5  |-  ( z  =  y  ->  (
z  e.  [_ z  /  x ]_ A  <->  y  e.  B ) )
34 dfsbcq 2993 . . . . . 6  |-  ( z  =  y  ->  ( [. z  /  x ]. ph  <->  [. y  /  x ]. ph ) )
35 sbsbc 2995 . . . . . . 7  |-  ( [ y  /  x ] ph 
<-> 
[. y  /  x ]. ph )
36 cbvralcsf.4 . . . . . . . 8  |-  F/ x ps
37 cbvralcsf.6 . . . . . . . 8  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
3836, 37sbie 1978 . . . . . . 7  |-  ( [ y  /  x ] ph 
<->  ps )
3935, 38bitr3i 242 . . . . . 6  |-  ( [. y  /  x ]. ph  <->  ps )
4034, 39syl6bb 252 . . . . 5  |-  ( z  =  y  ->  ( [. z  /  x ]. ph  <->  ps ) )
4133, 40imbi12d 311 . . . 4  |-  ( z  =  y  ->  (
( z  e.  [_ z  /  x ]_ A  ->  [. z  /  x ]. ph )  <->  ( y  e.  B  ->  ps )
) )
4218, 19, 41cbval 1924 . . 3  |-  ( A. z ( z  e. 
[_ z  /  x ]_ A  ->  [. z  /  x ]. ph )  <->  A. y ( y  e.  B  ->  ps )
)
4311, 42bitri 240 . 2  |-  ( A. x ( x  e.  A  ->  ph )  <->  A. y
( y  e.  B  ->  ps ) )
44 df-ral 2548 . 2  |-  ( A. x  e.  A  ph  <->  A. x
( x  e.  A  ->  ph ) )
45 df-ral 2548 . 2  |-  ( A. y  e.  B  ps  <->  A. y ( y  e.  B  ->  ps )
)
4643, 44, 453bitr4i 268 1  |-  ( A. x  e.  A  ph  <->  A. y  e.  B  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   A.wal 1527   F/wnf 1531    = wceq 1623   [wsb 1629    e. wcel 1684   {cab 2269   F/_wnfc 2406   A.wral 2543   [.wsbc 2991   [_csb 3081
This theorem is referenced by:  cbvrexcsf  3144  cbvralv2  3147
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-sbc 2992  df-csb 3082
  Copyright terms: Public domain W3C validator