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Theorem exss 4018
Description: Restricted existence in a class (even if proper) implies restricted existence in a subset. (Contributed by NM, 23-Aug-2003.)
Assertion
Ref Expression
exss  |-  ( E. x  e.  A  ph  ->  E. y ( y 
C_  A  /\  E. x  e.  y  ph ) )
Distinct variable groups:    x, y, A    ph, y
Allowed substitution hint:    ph( x)

Proof of Theorem exss
Dummy variables  z  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rabn0m 3293 . . 3  |-  ( E. z  z  e.  {
x  e.  A  |  ph }  <->  E. x  e.  A  ph )
2 df-rab 2362 . . . . 5  |-  { x  e.  A  |  ph }  =  { x  |  ( x  e.  A  /\  ph ) }
32eleq2i 2149 . . . 4  |-  ( z  e.  { x  e.  A  |  ph }  <->  z  e.  { x  |  ( x  e.  A  /\  ph ) } )
43exbii 1537 . . 3  |-  ( E. z  z  e.  {
x  e.  A  |  ph }  <->  E. z  z  e. 
{ x  |  ( x  e.  A  /\  ph ) } )
51, 4bitr3i 184 . 2  |-  ( E. x  e.  A  ph  <->  E. z  z  e.  {
x  |  ( x  e.  A  /\  ph ) } )
6 vex 2615 . . . . . 6  |-  z  e. 
_V
76snss 3540 . . . . 5  |-  ( z  e.  { x  |  ( x  e.  A  /\  ph ) }  <->  { z }  C_  { x  |  ( x  e.  A  /\  ph ) } )
8 ssab2 3089 . . . . . 6  |-  { x  |  ( x  e.  A  /\  ph ) }  C_  A
9 sstr2 3017 . . . . . 6  |-  ( { z }  C_  { x  |  ( x  e.  A  /\  ph ) }  ->  ( { x  |  ( x  e.  A  /\  ph ) }  C_  A  ->  { z }  C_  A )
)
108, 9mpi 15 . . . . 5  |-  ( { z }  C_  { x  |  ( x  e.  A  /\  ph ) }  ->  { z } 
C_  A )
117, 10sylbi 119 . . . 4  |-  ( z  e.  { x  |  ( x  e.  A  /\  ph ) }  ->  { z }  C_  A
)
12 simpr 108 . . . . . . . 8  |-  ( ( [ z  /  x ] x  e.  A  /\  [ z  /  x ] ph )  ->  [ z  /  x ] ph )
13 equsb1 1710 . . . . . . . . 9  |-  [ z  /  x ] x  =  z
14 velsn 3439 . . . . . . . . . 10  |-  ( x  e.  { z }  <-> 
x  =  z )
1514sbbii 1690 . . . . . . . . 9  |-  ( [ z  /  x ]
x  e.  { z }  <->  [ z  /  x ] x  =  z
)
1613, 15mpbir 144 . . . . . . . 8  |-  [ z  /  x ] x  e.  { z }
1712, 16jctil 305 . . . . . . 7  |-  ( ( [ z  /  x ] x  e.  A  /\  [ z  /  x ] ph )  ->  ( [ z  /  x ] x  e.  { z }  /\  [ z  /  x ] ph ) )
18 df-clab 2070 . . . . . . . 8  |-  ( z  e.  { x  |  ( x  e.  A  /\  ph ) }  <->  [ z  /  x ] ( x  e.  A  /\  ph ) )
19 sban 1872 . . . . . . . 8  |-  ( [ z  /  x ]
( x  e.  A  /\  ph )  <->  ( [
z  /  x ]
x  e.  A  /\  [ z  /  x ] ph ) )
2018, 19bitri 182 . . . . . . 7  |-  ( z  e.  { x  |  ( x  e.  A  /\  ph ) }  <->  ( [
z  /  x ]
x  e.  A  /\  [ z  /  x ] ph ) )
21 df-rab 2362 . . . . . . . . 9  |-  { x  e.  { z }  |  ph }  =  { x  |  ( x  e. 
{ z }  /\  ph ) }
2221eleq2i 2149 . . . . . . . 8  |-  ( z  e.  { x  e. 
{ z }  |  ph }  <->  z  e.  {
x  |  ( x  e.  { z }  /\  ph ) } )
23 df-clab 2070 . . . . . . . . 9  |-  ( z  e.  { x  |  ( x  e.  {
z }  /\  ph ) }  <->  [ z  /  x ] ( x  e. 
{ z }  /\  ph ) )
24 sban 1872 . . . . . . . . 9  |-  ( [ z  /  x ]
( x  e.  {
z }  /\  ph ) 
<->  ( [ z  /  x ] x  e.  {
z }  /\  [
z  /  x ] ph ) )
2523, 24bitri 182 . . . . . . . 8  |-  ( z  e.  { x  |  ( x  e.  {
z }  /\  ph ) }  <->  ( [ z  /  x ] x  e.  { z }  /\  [ z  /  x ] ph ) )
2622, 25bitri 182 . . . . . . 7  |-  ( z  e.  { x  e. 
{ z }  |  ph }  <->  ( [ z  /  x ] x  e.  { z }  /\  [ z  /  x ] ph ) )
2717, 20, 263imtr4i 199 . . . . . 6  |-  ( z  e.  { x  |  ( x  e.  A  /\  ph ) }  ->  z  e.  { x  e. 
{ z }  |  ph } )
28 elex2 2626 . . . . . 6  |-  ( z  e.  { x  e. 
{ z }  |  ph }  ->  E. w  w  e.  { x  e.  { z }  |  ph } )
2927, 28syl 14 . . . . 5  |-  ( z  e.  { x  |  ( x  e.  A  /\  ph ) }  ->  E. w  w  e.  {
x  e.  { z }  |  ph }
)
30 rabn0m 3293 . . . . 5  |-  ( E. w  w  e.  {
x  e.  { z }  |  ph }  <->  E. x  e.  { z } ph )
3129, 30sylib 120 . . . 4  |-  ( z  e.  { x  |  ( x  e.  A  /\  ph ) }  ->  E. x  e.  { z } ph )
326snex 3984 . . . . 5  |-  { z }  e.  _V
33 sseq1 3031 . . . . . 6  |-  ( y  =  { z }  ->  ( y  C_  A 
<->  { z }  C_  A ) )
34 rexeq 2556 . . . . . 6  |-  ( y  =  { z }  ->  ( E. x  e.  y  ph  <->  E. x  e.  { z } ph ) )
3533, 34anbi12d 457 . . . . 5  |-  ( y  =  { z }  ->  ( ( y 
C_  A  /\  E. x  e.  y  ph ) 
<->  ( { z } 
C_  A  /\  E. x  e.  { z } ph ) ) )
3632, 35spcev 2703 . . . 4  |-  ( ( { z }  C_  A  /\  E. x  e. 
{ z } ph )  ->  E. y ( y 
C_  A  /\  E. x  e.  y  ph ) )
3711, 31, 36syl2anc 403 . . 3  |-  ( z  e.  { x  |  ( x  e.  A  /\  ph ) }  ->  E. y ( y  C_  A  /\  E. x  e.  y  ph ) )
3837exlimiv 1530 . 2  |-  ( E. z  z  e.  {
x  |  ( x  e.  A  /\  ph ) }  ->  E. y
( y  C_  A  /\  E. x  e.  y 
ph ) )
395, 38sylbi 119 1  |-  ( E. x  e.  A  ph  ->  E. y ( y 
C_  A  /\  E. x  e.  y  ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1285   E.wex 1422    e. wcel 1434   [wsb 1687   {cab 2069   E.wrex 2354   {crab 2357    C_ wss 2984   {csn 3422
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-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-pow 3974
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-rex 2359  df-rab 2362  df-v 2614  df-in 2990  df-ss 2997  df-pw 3408  df-sn 3428
This theorem is referenced by: (None)
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