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Theorem elabgft1 10304
Description: One implication of elabgf 2708, 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 bi1 115 . . . . . 6  |-  ( ( A  e.  { x  |  ph }  <->  ph )  -> 
( A  e.  {
x  |  ph }  ->  ph ) )
2 imim2 53 . . . . . 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 1360 . . 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 2196 . . . . . 6  |-  F/_ x { x  |  ph }
86, 7nfel 2202 . . . . 5  |-  F/ x  A  e.  { x  |  ph }
9 elabgf1.nf2 . . . . 5  |-  F/ x ps
108, 9nfim 1480 . . . 4  |-  F/ x
( A  e.  {
x  |  ph }  ->  ps )
11 elabgf0 10303 . . . 4  |-  ( x  =  A  ->  ( A  e.  { x  |  ph }  <->  ph ) )
126, 10, 11bj-vtoclgft 10301 . . 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 48 1  |-  ( A. x ( x  =  A  ->  ( ph  ->  ps ) )  -> 
( A  e.  {
x  |  ph }  ->  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 102   A.wal 1257    = wceq 1259   F/wnf 1365    e. wcel 1409   {cab 2042   F/_wnfc 2181
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576
This theorem is referenced by:  elabgf1  10305
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