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Theorem rgen2a 2392
Description: Generalization rule for restricted quantification. Note that  x and  y needn't be distinct (and illustrates the use of dvelimor 1910). (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 1437 . . . . 5  |-  F/ y  z  e.  A
2 eleq1 2116 . . . . 5  |-  ( z  =  x  ->  (
z  e.  A  <->  x  e.  A ) )
31, 2dvelimor 1910 . . . 4  |-  ( A. y  y  =  x  \/  F/ y  x  e.  A )
4 eleq1 2116 . . . . . . . . 9  |-  ( y  =  x  ->  (
y  e.  A  <->  x  e.  A ) )
5 rgen2a.1 . . . . . . . . . 10  |-  ( ( x  e.  A  /\  y  e.  A )  ->  ph )
65ex 112 . . . . . . . . 9  |-  ( x  e.  A  ->  (
y  e.  A  ->  ph ) )
74, 6syl6bi 156 . . . . . . . 8  |-  ( y  =  x  ->  (
y  e.  A  -> 
( y  e.  A  ->  ph ) ) )
87pm2.43d 48 . . . . . . 7  |-  ( y  =  x  ->  (
y  e.  A  ->  ph ) )
98alimi 1360 . . . . . 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 1427 . . . . . 6  |-  ( F/ y  x  e.  A  ->  ( x  e.  A  ->  A. y  x  e.  A ) )
126alimi 1360 . . . . . 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 646 . . . 4  |-  ( ( A. y  y  =  x  \/  F/ y  x  e.  A )  ->  ( x  e.  A  ->  A. y
( y  e.  A  ->  ph ) ) )
153, 14ax-mp 7 . . 3  |-  ( x  e.  A  ->  A. y
( y  e.  A  ->  ph ) )
16 df-ral 2328 . . 3  |-  ( A. y  e.  A  ph  <->  A. y
( y  e.  A  ->  ph ) )
1715, 16sylibr 141 . 2  |-  ( x  e.  A  ->  A. y  e.  A  ph )
1817rgen 2391 1  |-  A. x  e.  A  A. y  e.  A  ph
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    \/ wo 639   A.wal 1257    = wceq 1259   F/wnf 1365    e. wcel 1409   A.wral 2323
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-nf 1366  df-sb 1662  df-cleq 2049  df-clel 2052  df-ral 2328
This theorem is referenced by:  ordsucunielexmid  4284  onintexmid  4325  isoid  5478  issmo  5934  ecopover  6235  ecopoverg  6238  subf  7276  cnref1o  8680  ioof  8941  fzof  9103
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