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Theorem rgen2a 2461
Description: Generalization rule for restricted quantification. Note that  x and  y needn't be distinct (and illustrates the use of dvelimor 1969). (Contributed by NM, 23-Nov-1994.) (Proof rewritten by Jim Kingdon, 1-Jun-2018.)
Hypothesis
Ref Expression
rgen2a.1  |-  ( ( x  e.  A  /\  y  e.  A )  ->  ph )
Assertion
Ref Expression
rgen2a  |-  A. x  e.  A  A. y  e.  A  ph
Distinct variable group:    y, A
Allowed substitution hints:    ph( x, y)    A( x)

Proof of Theorem rgen2a
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 nfv 1491 . . . . 5  |-  F/ y  z  e.  A
2 eleq1 2178 . . . . 5  |-  ( z  =  x  ->  (
z  e.  A  <->  x  e.  A ) )
31, 2dvelimor 1969 . . . 4  |-  ( A. y  y  =  x  \/  F/ y  x  e.  A )
4 eleq1 2178 . . . . . . . . 9  |-  ( y  =  x  ->  (
y  e.  A  <->  x  e.  A ) )
5 rgen2a.1 . . . . . . . . . 10  |-  ( ( x  e.  A  /\  y  e.  A )  ->  ph )
65ex 114 . . . . . . . . 9  |-  ( x  e.  A  ->  (
y  e.  A  ->  ph ) )
74, 6syl6bi 162 . . . . . . . 8  |-  ( y  =  x  ->  (
y  e.  A  -> 
( y  e.  A  ->  ph ) ) )
87pm2.43d 50 . . . . . . 7  |-  ( y  =  x  ->  (
y  e.  A  ->  ph ) )
98alimi 1414 . . . . . 6  |-  ( A. y  y  =  x  ->  A. y ( y  e.  A  ->  ph )
)
109a1d 22 . . . . 5  |-  ( A. y  y  =  x  ->  ( x  e.  A  ->  A. y ( y  e.  A  ->  ph )
) )
11 nfr 1481 . . . . . 6  |-  ( F/ y  x  e.  A  ->  ( x  e.  A  ->  A. y  x  e.  A ) )
126alimi 1414 . . . . . 6  |-  ( A. y  x  e.  A  ->  A. y ( y  e.  A  ->  ph )
)
1311, 12syl6 33 . . . . 5  |-  ( F/ y  x  e.  A  ->  ( x  e.  A  ->  A. y ( y  e.  A  ->  ph )
) )
1410, 13jaoi 688 . . . 4  |-  ( ( A. y  y  =  x  \/  F/ y  x  e.  A )  ->  ( x  e.  A  ->  A. y
( y  e.  A  ->  ph ) ) )
153, 14ax-mp 5 . . 3  |-  ( x  e.  A  ->  A. y
( y  e.  A  ->  ph ) )
16 df-ral 2396 . . 3  |-  ( A. y  e.  A  ph  <->  A. y
( y  e.  A  ->  ph ) )
1715, 16sylibr 133 . 2  |-  ( x  e.  A  ->  A. y  e.  A  ph )
1817rgen 2460 1  |-  A. x  e.  A  A. y  e.  A  ph
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    \/ wo 680   A.wal 1312    = wceq 1314   F/wnf 1419    e. wcel 1463   A.wral 2391
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-nf 1420  df-sb 1719  df-cleq 2108  df-clel 2111  df-ral 2396
This theorem is referenced by:  ordsucunielexmid  4414  onintexmid  4455  isoid  5677  issmo  6151  oawordriexmid  6332  ecopover  6493  ecopoverg  6496  1domsn  6679  unfiexmid  6772  axaddf  7640  axmulf  7641  subf  7928  negiso  8670  cnref1o  9389  xaddf  9567  ioof  9694  fzof  9861  xrnegiso  10971  reeff1  11306  gcdf  11557  eucalgf  11632  qredeu  11674  qnnen  11839  strsetsid  11887  hmeofn  12366  ismeti  12410  qtopbasss  12585  tgqioo  12611  peano4nninf  13011
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