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Theorem elabgf2 11125
Description: One implication of elabgf 2749. (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 2227 . . . 4  |-  F/_ x { x  |  ph }
41, 3nfel 2233 . . 3  |-  F/ x  A  e.  { x  |  ph }
52, 4nfim 1507 . 2  |-  F/ x
( ps  ->  A  e.  { x  |  ph } )
6 elabgf0 11122 . 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 11121 1  |-  ( A  e.  B  ->  ( ps  ->  A  e.  {
x  |  ph }
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103    = wceq 1287   F/wnf 1392    e. wcel 1436   {cab 2071   F/_wnfc 2212
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 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-v 2617
This theorem is referenced by:  elabf2  11127  elabg2  11130  bj-intabssel1  11135
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