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Theorem elabgf2 13776
Description: One implication of elabgf 2872. (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 2314 . . . 4  |-  F/_ x { x  |  ph }
41, 3nfel 2321 . . 3  |-  F/ x  A  e.  { x  |  ph }
52, 4nfim 1565 . 2  |-  F/ x
( ps  ->  A  e.  { x  |  ph } )
6 elabgf0 13773 . 2  |-  ( x  =  A  ->  ( A  e.  { x  |  ph }  <->  ph ) )
7 bicom1 130 . . 3  |-  ( ( A  e.  { x  |  ph }  <->  ph )  -> 
( ph  <->  A  e.  { x  |  ph } ) )
8 elabgf2.1 . . . 4  |-  ( x  =  A  ->  ( ps  ->  ph ) )
9 biimp 117 . . . 4  |-  ( (
ph 
<->  A  e.  { x  |  ph } )  -> 
( ph  ->  A  e. 
{ x  |  ph } ) )
108, 9syl9 72 . . 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 13772 1  |-  ( A  e.  B  ->  ( ps  ->  A  e.  {
x  |  ph }
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    = wceq 1348   F/wnf 1453    e. wcel 2141   {cab 2156   F/_wnfc 2299
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-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732
This theorem is referenced by:  elabf2  13778  elabg2  13781  bj-intabssel1  13786
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