ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  cbvrexcsf Unicode version

Theorem cbvrexcsf 2966
Description: A more general version of cbvrexf 2573 that has no distinct variable restrictions. Changes bound variables using implicit substitution. (Contributed by Andrew Salmon, 13-Jul-2011.) (Proof shortened by Mario Carneiro, 7-Dec-2014.)
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
cbvrexcsf  |-  ( E. x  e.  A  ph  <->  E. y  e.  B  ps )

Proof of Theorem cbvrexcsf
Dummy variables  v  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfv 1462 . . . 4  |-  F/ z ( x  e.  A  /\  ph )
2 nfcsb1v 2939 . . . . . 6  |-  F/_ x [_ z  /  x ]_ A
32nfcri 2214 . . . . 5  |-  F/ x  z  e.  [_ z  /  x ]_ A
4 nfsbc1v 2834 . . . . 5  |-  F/ x [. z  /  x ]. ph
53, 4nfan 1498 . . . 4  |-  F/ x
( z  e.  [_ z  /  x ]_ A  /\  [. z  /  x ]. ph )
6 id 19 . . . . . 6  |-  ( x  =  z  ->  x  =  z )
7 csbeq1a 2917 . . . . . 6  |-  ( x  =  z  ->  A  =  [_ z  /  x ]_ A )
86, 7eleq12d 2150 . . . . 5  |-  ( x  =  z  ->  (
x  e.  A  <->  z  e.  [_ z  /  x ]_ A ) )
9 sbceq1a 2825 . . . . 5  |-  ( x  =  z  ->  ( ph 
<-> 
[. z  /  x ]. ph ) )
108, 9anbi12d 457 . . . 4  |-  ( x  =  z  ->  (
( x  e.  A  /\  ph )  <->  ( z  e.  [_ z  /  x ]_ A  /\  [. z  /  x ]. ph )
) )
111, 5, 10cbvex 1680 . . 3  |-  ( E. x ( x  e.  A  /\  ph )  <->  E. z ( z  e. 
[_ z  /  x ]_ A  /\  [. z  /  x ]. ph )
)
12 nfcv 2220 . . . . . . 7  |-  F/_ y
z
13 cbvralcsf.1 . . . . . . 7  |-  F/_ y A
1412, 13nfcsb 2941 . . . . . 6  |-  F/_ y [_ z  /  x ]_ A
1514nfcri 2214 . . . . 5  |-  F/ y  z  e.  [_ z  /  x ]_ A
16 cbvralcsf.3 . . . . . 6  |-  F/ y
ph
1712, 16nfsbc 2836 . . . . 5  |-  F/ y
[. z  /  x ]. ph
1815, 17nfan 1498 . . . 4  |-  F/ y ( z  e.  [_ z  /  x ]_ A  /\  [. z  /  x ]. ph )
19 nfv 1462 . . . 4  |-  F/ z ( y  e.  B  /\  ps )
20 id 19 . . . . . 6  |-  ( z  =  y  ->  z  =  y )
21 csbeq1 2912 . . . . . . 7  |-  ( z  =  y  ->  [_ z  /  x ]_ A  = 
[_ y  /  x ]_ A )
22 df-csb 2910 . . . . . . . 8  |-  [_ y  /  x ]_ A  =  { v  |  [. y  /  x ]. v  e.  A }
23 cbvralcsf.2 . . . . . . . . . . . 12  |-  F/_ x B
2423nfcri 2214 . . . . . . . . . . 11  |-  F/ x  v  e.  B
25 cbvralcsf.5 . . . . . . . . . . . 12  |-  ( x  =  y  ->  A  =  B )
2625eleq2d 2149 . . . . . . . . . . 11  |-  ( x  =  y  ->  (
v  e.  A  <->  v  e.  B ) )
2724, 26sbie 1715 . . . . . . . . . 10  |-  ( [ y  /  x ]
v  e.  A  <->  v  e.  B )
28 sbsbc 2820 . . . . . . . . . 10  |-  ( [ y  /  x ]
v  e.  A  <->  [. y  /  x ]. v  e.  A
)
2927, 28bitr3i 184 . . . . . . . . 9  |-  ( v  e.  B  <->  [. y  /  x ]. v  e.  A
)
3029abbi2i 2194 . . . . . . . 8  |-  B  =  { v  |  [. y  /  x ]. v  e.  A }
3122, 30eqtr4i 2105 . . . . . . 7  |-  [_ y  /  x ]_ A  =  B
3221, 31syl6eq 2130 . . . . . 6  |-  ( z  =  y  ->  [_ z  /  x ]_ A  =  B )
3320, 32eleq12d 2150 . . . . 5  |-  ( z  =  y  ->  (
z  e.  [_ z  /  x ]_ A  <->  y  e.  B ) )
34 dfsbcq 2818 . . . . . 6  |-  ( z  =  y  ->  ( [. z  /  x ]. ph  <->  [. y  /  x ]. ph ) )
35 sbsbc 2820 . . . . . . 7  |-  ( [ y  /  x ] ph 
<-> 
[. y  /  x ]. ph )
36 cbvralcsf.4 . . . . . . . 8  |-  F/ x ps
37 cbvralcsf.6 . . . . . . . 8  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
3836, 37sbie 1715 . . . . . . 7  |-  ( [ y  /  x ] ph 
<->  ps )
3935, 38bitr3i 184 . . . . . 6  |-  ( [. y  /  x ]. ph  <->  ps )
4034, 39syl6bb 194 . . . . 5  |-  ( z  =  y  ->  ( [. z  /  x ]. ph  <->  ps ) )
4133, 40anbi12d 457 . . . 4  |-  ( z  =  y  ->  (
( z  e.  [_ z  /  x ]_ A  /\  [. z  /  x ]. ph )  <->  ( y  e.  B  /\  ps )
) )
4218, 19, 41cbvex 1680 . . 3  |-  ( E. z ( z  e. 
[_ z  /  x ]_ A  /\  [. z  /  x ]. ph )  <->  E. y ( y  e.  B  /\  ps )
)
4311, 42bitri 182 . 2  |-  ( E. x ( x  e.  A  /\  ph )  <->  E. y ( y  e.  B  /\  ps )
)
44 df-rex 2355 . 2  |-  ( E. x  e.  A  ph  <->  E. x ( x  e.  A  /\  ph )
)
45 df-rex 2355 . 2  |-  ( E. y  e.  B  ps  <->  E. y ( y  e.  B  /\  ps )
)
4643, 44, 453bitr4i 210 1  |-  ( E. x  e.  A  ph  <->  E. y  e.  B  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1285   F/wnf 1390   E.wex 1422    e. wcel 1434   [wsb 1686   {cab 2068   F/_wnfc 2207   E.wrex 2350   [.wsbc 2816   [_csb 2909
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-rex 2355  df-sbc 2817  df-csb 2910
This theorem is referenced by:  cbvrexv2  2970
  Copyright terms: Public domain W3C validator