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Theorem rgen2a 2369
Description: Generalization rule for restricted quantification. Note that x and y needn't be distinct (and illustrates the use of dvelimor 1891). (Contributed by NM, 23-Nov-1994.) (Proof rewritten by Jim Kingdon, 1-Jun-2018.)
Hypothesis
Ref Expression
rgen2a.1 ((x A y A) → φ)
Assertion
Ref Expression
rgen2a x A y A φ
Distinct variable group:   y,A
Allowed substitution hints:   φ(x,y)   A(x)

Proof of Theorem rgen2a
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 nfv 1418 . . . . 5 y z A
2 eleq1 2097 . . . . 5 (z = x → (z Ax A))
31, 2dvelimor 1891 . . . 4 (y y = x y x A)
4 eleq1 2097 . . . . . . . . 9 (y = x → (y Ax A))
5 rgen2a.1 . . . . . . . . . 10 ((x A y A) → φ)
65ex 108 . . . . . . . . 9 (x A → (y Aφ))
74, 6syl6bi 152 . . . . . . . 8 (y = x → (y A → (y Aφ)))
87pm2.43d 44 . . . . . . 7 (y = x → (y Aφ))
98alimi 1341 . . . . . 6 (y y = xy(y Aφ))
109a1d 22 . . . . 5 (y y = x → (x Ay(y Aφ)))
11 nfr 1408 . . . . . 6 (Ⅎy x A → (x Ay x A))
126alimi 1341 . . . . . 6 (y x Ay(y Aφ))
1311, 12syl6 29 . . . . 5 (Ⅎy x A → (x Ay(y Aφ)))
1410, 13jaoi 635 . . . 4 ((y y = x y x A) → (x Ay(y Aφ)))
153, 14ax-mp 7 . . 3 (x Ay(y Aφ))
16 df-ral 2305 . . 3 (y A φy(y Aφ))
1715, 16sylibr 137 . 2 (x Ay A φ)
1817rgen 2368 1 x A y A φ
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   wo 628  wal 1240   = wceq 1242  wnf 1346   wcel 1390  wral 2300
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-cleq 2030  df-clel 2033  df-ral 2305
This theorem is referenced by:  ordsucunielexmid  4216  isoid  5393  issmo  5844  ecopover  6140  ecopoverg  6143  subf  6970  cnref1o  8317  ioof  8570  fzof  8731
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