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Theorem abeq0 3363
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 1903 . . 3  |-  ( [ y  /  x ]  -.  ph  <->  -.  [ y  /  x ] ph )
21albii 1431 . 2  |-  ( A. y [ y  /  x ]  -.  ph  <->  A. y  -.  [
y  /  x ] ph )
3 nfv 1493 . . 3  |-  F/ y  -.  ph
43sb8 1812 . 2  |-  ( A. x  -.  ph  <->  A. y [ y  /  x ]  -.  ph )
5 eq0 3351 . . 3  |-  ( { x  |  ph }  =  (/)  <->  A. y  -.  y  e.  { x  |  ph } )
6 df-clab 2104 . . . . 5  |-  ( y  e.  { x  | 
ph }  <->  [ y  /  x ] ph )
76notbii 642 . . . 4  |-  ( -.  y  e.  { x  |  ph }  <->  -.  [ y  /  x ] ph )
87albii 1431 . . 3  |-  ( A. y  -.  y  e.  {
x  |  ph }  <->  A. y  -.  [ y  /  x ] ph )
95, 8bitri 183 . 2  |-  ( { x  |  ph }  =  (/)  <->  A. y  -.  [
y  /  x ] ph )
102, 4, 93bitr4ri 212 1  |-  ( { x  |  ph }  =  (/)  <->  A. x  -.  ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 104   A.wal 1314    = wceq 1316    e. wcel 1465   [wsb 1720   {cab 2103   (/)c0 3333
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-dif 3043  df-nul 3334
This theorem is referenced by:  opprc  3696
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