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Theorem elabgft1 15215
Description: One implication of elabgf 2902, 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 1466 . . 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 2338 . . . . . 6  |-  F/_ x { x  |  ph }
86, 7nfel 2345 . . . . 5  |-  F/ x  A  e.  { x  |  ph }
9 elabgf1.nf2 . . . . 5  |-  F/ x ps
108, 9nfim 1583 . . . 4  |-  F/ x
( A  e.  {
x  |  ph }  ->  ps )
11 elabgf0 15214 . . . 4  |-  ( x  =  A  ->  ( A  e.  { x  |  ph }  <->  ph ) )
126, 10, 11bj-vtoclgft 15212 . . 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 1362    = wceq 1364   F/wnf 1471    e. wcel 2164   {cab 2179   F/_wnfc 2323
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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762
This theorem is referenced by:  elabgf1  15216
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