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Theorem exss 3990
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 3272 . . 3  |-  ( E. z  z  e.  {
x  e.  A  |  ph }  <->  E. x  e.  A  ph )
2 df-rab 2332 . . . . 5  |-  { x  e.  A  |  ph }  =  { x  |  ( x  e.  A  /\  ph ) }
32eleq2i 2120 . . . 4  |-  ( z  e.  { x  e.  A  |  ph }  <->  z  e.  { x  |  ( x  e.  A  /\  ph ) } )
43exbii 1512 . . 3  |-  ( E. z  z  e.  {
x  e.  A  |  ph }  <->  E. z  z  e. 
{ x  |  ( x  e.  A  /\  ph ) } )
51, 4bitr3i 179 . 2  |-  ( E. x  e.  A  ph  <->  E. z  z  e.  {
x  |  ( x  e.  A  /\  ph ) } )
6 vex 2577 . . . . . 6  |-  z  e. 
_V
76snss 3521 . . . . 5  |-  ( z  e.  { x  |  ( x  e.  A  /\  ph ) }  <->  { z }  C_  { x  |  ( x  e.  A  /\  ph ) } )
8 ssab2 3051 . . . . . 6  |-  { x  |  ( x  e.  A  /\  ph ) }  C_  A
9 sstr2 2979 . . . . . 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 118 . . . 4  |-  ( z  e.  { x  |  ( x  e.  A  /\  ph ) }  ->  { z }  C_  A
)
12 simpr 107 . . . . . . . 8  |-  ( ( [ z  /  x ] x  e.  A  /\  [ z  /  x ] ph )  ->  [ z  /  x ] ph )
13 equsb1 1684 . . . . . . . . 9  |-  [ z  /  x ] x  =  z
14 velsn 3419 . . . . . . . . . 10  |-  ( x  e.  { z }  <-> 
x  =  z )
1514sbbii 1664 . . . . . . . . 9  |-  ( [ z  /  x ]
x  e.  { z }  <->  [ z  /  x ] x  =  z
)
1613, 15mpbir 138 . . . . . . . 8  |-  [ z  /  x ] x  e.  { z }
1712, 16jctil 299 . . . . . . 7  |-  ( ( [ z  /  x ] x  e.  A  /\  [ z  /  x ] ph )  ->  ( [ z  /  x ] x  e.  { z }  /\  [ z  /  x ] ph ) )
18 df-clab 2043 . . . . . . . 8  |-  ( z  e.  { x  |  ( x  e.  A  /\  ph ) }  <->  [ z  /  x ] ( x  e.  A  /\  ph ) )
19 sban 1845 . . . . . . . 8  |-  ( [ z  /  x ]
( x  e.  A  /\  ph )  <->  ( [
z  /  x ]
x  e.  A  /\  [ z  /  x ] ph ) )
2018, 19bitri 177 . . . . . . 7  |-  ( z  e.  { x  |  ( x  e.  A  /\  ph ) }  <->  ( [
z  /  x ]
x  e.  A  /\  [ z  /  x ] ph ) )
21 df-rab 2332 . . . . . . . . 9  |-  { x  e.  { z }  |  ph }  =  { x  |  ( x  e. 
{ z }  /\  ph ) }
2221eleq2i 2120 . . . . . . . 8  |-  ( z  e.  { x  e. 
{ z }  |  ph }  <->  z  e.  {
x  |  ( x  e.  { z }  /\  ph ) } )
23 df-clab 2043 . . . . . . . . 9  |-  ( z  e.  { x  |  ( x  e.  {
z }  /\  ph ) }  <->  [ z  /  x ] ( x  e. 
{ z }  /\  ph ) )
24 sban 1845 . . . . . . . . 9  |-  ( [ z  /  x ]
( x  e.  {
z }  /\  ph ) 
<->  ( [ z  /  x ] x  e.  {
z }  /\  [
z  /  x ] ph ) )
2523, 24bitri 177 . . . . . . . 8  |-  ( z  e.  { x  |  ( x  e.  {
z }  /\  ph ) }  <->  ( [ z  /  x ] x  e.  { z }  /\  [ z  /  x ] ph ) )
2622, 25bitri 177 . . . . . . 7  |-  ( z  e.  { x  e. 
{ z }  |  ph }  <->  ( [ z  /  x ] x  e.  { z }  /\  [ z  /  x ] ph ) )
2717, 20, 263imtr4i 194 . . . . . 6  |-  ( z  e.  { x  |  ( x  e.  A  /\  ph ) }  ->  z  e.  { x  e. 
{ z }  |  ph } )
28 elex2 2587 . . . . . 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 3272 . . . . 5  |-  ( E. w  w  e.  {
x  e.  { z }  |  ph }  <->  E. x  e.  { z } ph )
3129, 30sylib 131 . . . 4  |-  ( z  e.  { x  |  ( x  e.  A  /\  ph ) }  ->  E. x  e.  { z } ph )
32 snexgOLD 3962 . . . . . 6  |-  ( z  e.  _V  ->  { z }  e.  _V )
336, 32ax-mp 7 . . . . 5  |-  { z }  e.  _V
34 sseq1 2993 . . . . . 6  |-  ( y  =  { z }  ->  ( y  C_  A 
<->  { z }  C_  A ) )
35 rexeq 2523 . . . . . 6  |-  ( y  =  { z }  ->  ( E. x  e.  y  ph  <->  E. x  e.  { z } ph ) )
3634, 35anbi12d 450 . . . . 5  |-  ( y  =  { z }  ->  ( ( y 
C_  A  /\  E. x  e.  y  ph ) 
<->  ( { z } 
C_  A  /\  E. x  e.  { z } ph ) ) )
3733, 36spcev 2664 . . . 4  |-  ( ( { z }  C_  A  /\  E. x  e. 
{ z } ph )  ->  E. y ( y 
C_  A  /\  E. x  e.  y  ph ) )
3811, 31, 37syl2anc 397 . . 3  |-  ( z  e.  { x  |  ( x  e.  A  /\  ph ) }  ->  E. y ( y  C_  A  /\  E. x  e.  y  ph ) )
3938exlimiv 1505 . 2  |-  ( E. z  z  e.  {
x  |  ( x  e.  A  /\  ph ) }  ->  E. y
( y  C_  A  /\  E. x  e.  y 
ph ) )
405, 39sylbi 118 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 101    = wceq 1259   E.wex 1397    e. wcel 1409   [wsb 1661   {cab 2042   E.wrex 2324   {crab 2327   _Vcvv 2574    C_ wss 2944   {csn 3402
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-rex 2329  df-rab 2332  df-v 2576  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408
This theorem is referenced by: (None)
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