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Theorem rgen2a 2463
Description: Generalization rule for restricted quantification. Note that  x and  y needn't be distinct (and illustrates the use of dvelimor 1971). (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 1493 . . . . 5  |-  F/ y  z  e.  A
2 eleq1 2180 . . . . 5  |-  ( z  =  x  ->  (
z  e.  A  <->  x  e.  A ) )
31, 2dvelimor 1971 . . . 4  |-  ( A. y  y  =  x  \/  F/ y  x  e.  A )
4 eleq1 2180 . . . . . . . . 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 1416 . . . . . 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 1483 . . . . . 6  |-  ( F/ y  x  e.  A  ->  ( x  e.  A  ->  A. y  x  e.  A ) )
126alimi 1416 . . . . . 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 690 . . . 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 2398 . . 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 2462 1  |-  A. x  e.  A  A. y  e.  A  ph
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    \/ wo 682   A.wal 1314    = wceq 1316   F/wnf 1421    e. wcel 1465   A.wral 2393
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-nf 1422  df-sb 1721  df-cleq 2110  df-clel 2113  df-ral 2398
This theorem is referenced by:  ordsucunielexmid  4416  onintexmid  4457  isoid  5679  issmo  6153  oawordriexmid  6334  ecopover  6495  ecopoverg  6498  1domsn  6681  unfiexmid  6774  axaddf  7644  axmulf  7645  subf  7932  negiso  8681  cnref1o  9408  xaddf  9595  ioof  9722  fzof  9889  xrnegiso  10999  reeff1  11334  gcdf  11588  eucalgf  11663  qredeu  11705  qnnen  11871  strsetsid  11919  hmeofn  12398  ismeti  12442  qtopbasss  12617  tgqioo  12643  peano4nninf  13127
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