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Theorem elabgft1 14179
Description: One implication of elabgf 2879, in closed form. (Contributed by BJ, 21-Nov-2019.)
Hypotheses
Ref Expression
elabgf1.nf1  |-  F/_ x A
elabgf1.nf2  |-  F/ x ps
Assertion
Ref Expression
elabgft1  |-  ( A. x ( x  =  A  ->  ( ph  ->  ps ) )  -> 
( A  e.  {
x  |  ph }  ->  ps ) )

Proof of Theorem elabgft1
StepHypRef Expression
1 biimp 118 . . . . . 6  |-  ( ( A  e.  { x  |  ph }  <->  ph )  -> 
( A  e.  {
x  |  ph }  ->  ph ) )
2 imim2 55 . . . . . 6  |-  ( (
ph  ->  ps )  -> 
( ( A  e. 
{ x  |  ph }  ->  ph )  ->  ( A  e.  { x  |  ph }  ->  ps ) ) )
31, 2syl5 32 . . . . 5  |-  ( (
ph  ->  ps )  -> 
( ( A  e. 
{ x  |  ph } 
<-> 
ph )  ->  ( A  e.  { x  |  ph }  ->  ps ) ) )
43imim2i 12 . . . 4  |-  ( ( x  =  A  -> 
( ph  ->  ps )
)  ->  ( x  =  A  ->  ( ( A  e.  { x  |  ph }  <->  ph )  -> 
( A  e.  {
x  |  ph }  ->  ps ) ) ) )
54alimi 1455 . . 3  |-  ( A. x ( x  =  A  ->  ( ph  ->  ps ) )  ->  A. x ( x  =  A  ->  ( ( A  e.  { x  |  ph }  <->  ph )  -> 
( A  e.  {
x  |  ph }  ->  ps ) ) ) )
6 elabgf1.nf1 . . . 4  |-  F/_ x A
7 nfab1 2321 . . . . . 6  |-  F/_ x { x  |  ph }
86, 7nfel 2328 . . . . 5  |-  F/ x  A  e.  { x  |  ph }
9 elabgf1.nf2 . . . . 5  |-  F/ x ps
108, 9nfim 1572 . . . 4  |-  F/ x
( A  e.  {
x  |  ph }  ->  ps )
11 elabgf0 14178 . . . 4  |-  ( x  =  A  ->  ( A  e.  { x  |  ph }  <->  ph ) )
126, 10, 11bj-vtoclgft 14176 . . 3  |-  ( A. x ( x  =  A  ->  ( ( A  e.  { x  |  ph }  <->  ph )  -> 
( A  e.  {
x  |  ph }  ->  ps ) ) )  ->  ( A  e. 
{ x  |  ph }  ->  ( A  e. 
{ x  |  ph }  ->  ps ) ) )
135, 12syl 14 . 2  |-  ( A. x ( x  =  A  ->  ( ph  ->  ps ) )  -> 
( A  e.  {
x  |  ph }  ->  ( A  e.  {
x  |  ph }  ->  ps ) ) )
1413pm2.43d 50 1  |-  ( A. x ( x  =  A  ->  ( ph  ->  ps ) )  -> 
( A  e.  {
x  |  ph }  ->  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 105   A.wal 1351    = wceq 1353   F/wnf 1460    e. wcel 2148   {cab 2163   F/_wnfc 2306
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-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739
This theorem is referenced by:  elabgf1  14180
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