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Theorem abeq0 3468
Description: Condition for a class abstraction to be empty. (Contributed by Jim Kingdon, 12-Aug-2018.)
Assertion
Ref Expression
abeq0  |-  ( { x  |  ph }  =  (/)  <->  A. x  -.  ph )

Proof of Theorem abeq0
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 sbn 1964 . . 3  |-  ( [ y  /  x ]  -.  ph  <->  -.  [ y  /  x ] ph )
21albii 1481 . 2  |-  ( A. y [ y  /  x ]  -.  ph  <->  A. y  -.  [
y  /  x ] ph )
3 nfv 1539 . . 3  |-  F/ y  -.  ph
43sb8 1867 . 2  |-  ( A. x  -.  ph  <->  A. y [ y  /  x ]  -.  ph )
5 eq0 3456 . . 3  |-  ( { x  |  ph }  =  (/)  <->  A. y  -.  y  e.  { x  |  ph } )
6 df-clab 2176 . . . . 5  |-  ( y  e.  { x  | 
ph }  <->  [ y  /  x ] ph )
76notbii 669 . . . 4  |-  ( -.  y  e.  { x  |  ph }  <->  -.  [ y  /  x ] ph )
87albii 1481 . . 3  |-  ( A. y  -.  y  e.  {
x  |  ph }  <->  A. y  -.  [ y  /  x ] ph )
95, 8bitri 184 . 2  |-  ( { x  |  ph }  =  (/)  <->  A. y  -.  [
y  /  x ] ph )
102, 4, 93bitr4ri 213 1  |-  ( { x  |  ph }  =  (/)  <->  A. x  -.  ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 105   A.wal 1362    = wceq 1364   [wsb 1773    e. wcel 2160   {cab 2175   (/)c0 3437
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-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-dif 3146  df-nul 3438
This theorem is referenced by:  opprc  3814
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