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

Theorem cbvrexcsf 3120
Description: A more general version of cbvrexf 2697 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 1528 . . . 4  |-  F/ z ( x  e.  A  /\  ph )
2 nfcsb1v 3090 . . . . . 6  |-  F/_ x [_ z  /  x ]_ A
32nfcri 2313 . . . . 5  |-  F/ x  z  e.  [_ z  /  x ]_ A
4 nfsbc1v 2981 . . . . 5  |-  F/ x [. z  /  x ]. ph
53, 4nfan 1565 . . . 4  |-  F/ x
( z  e.  [_ z  /  x ]_ A  /\  [. z  /  x ]. ph )
6 id 19 . . . . . 6  |-  ( x  =  z  ->  x  =  z )
7 csbeq1a 3066 . . . . . 6  |-  ( x  =  z  ->  A  =  [_ z  /  x ]_ A )
86, 7eleq12d 2248 . . . . 5  |-  ( x  =  z  ->  (
x  e.  A  <->  z  e.  [_ z  /  x ]_ A ) )
9 sbceq1a 2972 . . . . 5  |-  ( x  =  z  ->  ( ph 
<-> 
[. z  /  x ]. ph ) )
108, 9anbi12d 473 . . . 4  |-  ( x  =  z  ->  (
( x  e.  A  /\  ph )  <->  ( z  e.  [_ z  /  x ]_ A  /\  [. z  /  x ]. ph )
) )
111, 5, 10cbvex 1756 . . 3  |-  ( E. x ( x  e.  A  /\  ph )  <->  E. z ( z  e. 
[_ z  /  x ]_ A  /\  [. z  /  x ]. ph )
)
12 nfcv 2319 . . . . . . 7  |-  F/_ y
z
13 cbvralcsf.1 . . . . . . 7  |-  F/_ y A
1412, 13nfcsb 3094 . . . . . 6  |-  F/_ y [_ z  /  x ]_ A
1514nfcri 2313 . . . . 5  |-  F/ y  z  e.  [_ z  /  x ]_ A
16 cbvralcsf.3 . . . . . 6  |-  F/ y
ph
1712, 16nfsbc 2983 . . . . 5  |-  F/ y
[. z  /  x ]. ph
1815, 17nfan 1565 . . . 4  |-  F/ y ( z  e.  [_ z  /  x ]_ A  /\  [. z  /  x ]. ph )
19 nfv 1528 . . . 4  |-  F/ z ( y  e.  B  /\  ps )
20 id 19 . . . . . 6  |-  ( z  =  y  ->  z  =  y )
21 csbeq1 3060 . . . . . . 7  |-  ( z  =  y  ->  [_ z  /  x ]_ A  = 
[_ y  /  x ]_ A )
22 df-csb 3058 . . . . . . . 8  |-  [_ y  /  x ]_ A  =  { v  |  [. y  /  x ]. v  e.  A }
23 cbvralcsf.2 . . . . . . . . . . . 12  |-  F/_ x B
2423nfcri 2313 . . . . . . . . . . 11  |-  F/ x  v  e.  B
25 cbvralcsf.5 . . . . . . . . . . . 12  |-  ( x  =  y  ->  A  =  B )
2625eleq2d 2247 . . . . . . . . . . 11  |-  ( x  =  y  ->  (
v  e.  A  <->  v  e.  B ) )
2724, 26sbie 1791 . . . . . . . . . 10  |-  ( [ y  /  x ]
v  e.  A  <->  v  e.  B )
28 sbsbc 2966 . . . . . . . . . 10  |-  ( [ y  /  x ]
v  e.  A  <->  [. y  /  x ]. v  e.  A
)
2927, 28bitr3i 186 . . . . . . . . 9  |-  ( v  e.  B  <->  [. y  /  x ]. v  e.  A
)
3029abbi2i 2292 . . . . . . . 8  |-  B  =  { v  |  [. y  /  x ]. v  e.  A }
3122, 30eqtr4i 2201 . . . . . . 7  |-  [_ y  /  x ]_ A  =  B
3221, 31eqtrdi 2226 . . . . . 6  |-  ( z  =  y  ->  [_ z  /  x ]_ A  =  B )
3320, 32eleq12d 2248 . . . . 5  |-  ( z  =  y  ->  (
z  e.  [_ z  /  x ]_ A  <->  y  e.  B ) )
34 dfsbcq 2964 . . . . . 6  |-  ( z  =  y  ->  ( [. z  /  x ]. ph  <->  [. y  /  x ]. ph ) )
35 sbsbc 2966 . . . . . . 7  |-  ( [ y  /  x ] ph 
<-> 
[. y  /  x ]. ph )
36 cbvralcsf.4 . . . . . . . 8  |-  F/ x ps
37 cbvralcsf.6 . . . . . . . 8  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
3836, 37sbie 1791 . . . . . . 7  |-  ( [ y  /  x ] ph 
<->  ps )
3935, 38bitr3i 186 . . . . . 6  |-  ( [. y  /  x ]. ph  <->  ps )
4034, 39bitrdi 196 . . . . 5  |-  ( z  =  y  ->  ( [. z  /  x ]. ph  <->  ps ) )
4133, 40anbi12d 473 . . . 4  |-  ( z  =  y  ->  (
( z  e.  [_ z  /  x ]_ A  /\  [. z  /  x ]. ph )  <->  ( y  e.  B  /\  ps )
) )
4218, 19, 41cbvex 1756 . . 3  |-  ( E. z ( z  e. 
[_ z  /  x ]_ A  /\  [. z  /  x ]. ph )  <->  E. y ( y  e.  B  /\  ps )
)
4311, 42bitri 184 . 2  |-  ( E. x ( x  e.  A  /\  ph )  <->  E. y ( y  e.  B  /\  ps )
)
44 df-rex 2461 . 2  |-  ( E. x  e.  A  ph  <->  E. x ( x  e.  A  /\  ph )
)
45 df-rex 2461 . 2  |-  ( E. y  e.  B  ps  <->  E. y ( y  e.  B  /\  ps )
)
4643, 44, 453bitr4i 212 1  |-  ( E. x  e.  A  ph  <->  E. y  e.  B  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1353   F/wnf 1460   E.wex 1492   [wsb 1762    e. wcel 2148   {cab 2163   F/_wnfc 2306   E.wrex 2456   [.wsbc 2962   [_csb 3057
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-sbc 2963  df-csb 3058
This theorem is referenced by:  cbvrexv2  3124
  Copyright terms: Public domain W3C validator