Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >   Mathboxes  >  elabgf2 Unicode version

Theorem elabgf2 10741
Description: One implication of elabgf 2737. (Contributed by BJ, 21-Nov-2019.)
Hypotheses
Ref Expression
elabgf2.nf1  |-  F/_ x A
elabgf2.nf2  |-  F/ x ps
elabgf2.1  |-  ( x  =  A  ->  ( ps  ->  ph ) )
Assertion
Ref Expression
elabgf2  |-  ( A  e.  B  ->  ( ps  ->  A  e.  {
x  |  ph }
) )

Proof of Theorem elabgf2
StepHypRef Expression
1 elabgf2.nf1 . 2  |-  F/_ x A
2 elabgf2.nf2 . . 3  |-  F/ x ps
3 nfab1 2222 . . . 4  |-  F/_ x { x  |  ph }
41, 3nfel 2228 . . 3  |-  F/ x  A  e.  { x  |  ph }
52, 4nfim 1505 . 2  |-  F/ x
( ps  ->  A  e.  { x  |  ph } )
6 elabgf0 10738 . 2  |-  ( x  =  A  ->  ( A  e.  { x  |  ph }  <->  ph ) )
7 bicom1 129 . . 3  |-  ( ( A  e.  { x  |  ph }  <->  ph )  -> 
( ph  <->  A  e.  { x  |  ph } ) )
8 elabgf2.1 . . . 4  |-  ( x  =  A  ->  ( ps  ->  ph ) )
9 bi1 116 . . . 4  |-  ( (
ph 
<->  A  e.  { x  |  ph } )  -> 
( ph  ->  A  e. 
{ x  |  ph } ) )
108, 9syl9 71 . . 3  |-  ( x  =  A  ->  (
( ph  <->  A  e.  { x  |  ph } )  -> 
( ps  ->  A  e.  { x  |  ph } ) ) )
117, 10syl5 32 . 2  |-  ( x  =  A  ->  (
( A  e.  {
x  |  ph }  <->  ph )  ->  ( ps  ->  A  e.  { x  |  ph } ) ) )
121, 5, 6, 11bj-vtoclgf 10737 1  |-  ( A  e.  B  ->  ( ps  ->  A  e.  {
x  |  ph }
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103    = wceq 1285   F/wnf 1390    e. wcel 1434   {cab 2068   F/_wnfc 2207
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-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604
This theorem is referenced by:  elabf2  10743  elabg2  10746  bj-intabssel1  10751
  Copyright terms: Public domain W3C validator