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Theorem rgen2a 2429
Description: Generalization rule for restricted quantification. Note that 𝑥 and 𝑦 needn't be distinct (and illustrates the use of dvelimor 1942). (Contributed by NM, 23-Nov-1994.) (Proof rewritten by Jim Kingdon, 1-Jun-2018.)
Hypothesis
Ref Expression
rgen2a.1 ((𝑥𝐴𝑦𝐴) → 𝜑)
Assertion
Ref Expression
rgen2a 𝑥𝐴𝑦𝐴 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 1466 . . . . 5 𝑦 𝑧𝐴
2 eleq1 2150 . . . . 5 (𝑧 = 𝑥 → (𝑧𝐴𝑥𝐴))
31, 2dvelimor 1942 . . . 4 (∀𝑦 𝑦 = 𝑥 ∨ Ⅎ𝑦 𝑥𝐴)
4 eleq1 2150 . . . . . . . . 9 (𝑦 = 𝑥 → (𝑦𝐴𝑥𝐴))
5 rgen2a.1 . . . . . . . . . 10 ((𝑥𝐴𝑦𝐴) → 𝜑)
65ex 113 . . . . . . . . 9 (𝑥𝐴 → (𝑦𝐴𝜑))
74, 6syl6bi 161 . . . . . . . 8 (𝑦 = 𝑥 → (𝑦𝐴 → (𝑦𝐴𝜑)))
87pm2.43d 49 . . . . . . 7 (𝑦 = 𝑥 → (𝑦𝐴𝜑))
98alimi 1389 . . . . . 6 (∀𝑦 𝑦 = 𝑥 → ∀𝑦(𝑦𝐴𝜑))
109a1d 22 . . . . 5 (∀𝑦 𝑦 = 𝑥 → (𝑥𝐴 → ∀𝑦(𝑦𝐴𝜑)))
11 nfr 1456 . . . . . 6 (Ⅎ𝑦 𝑥𝐴 → (𝑥𝐴 → ∀𝑦 𝑥𝐴))
126alimi 1389 . . . . . 6 (∀𝑦 𝑥𝐴 → ∀𝑦(𝑦𝐴𝜑))
1311, 12syl6 33 . . . . 5 (Ⅎ𝑦 𝑥𝐴 → (𝑥𝐴 → ∀𝑦(𝑦𝐴𝜑)))
1410, 13jaoi 671 . . . 4 ((∀𝑦 𝑦 = 𝑥 ∨ Ⅎ𝑦 𝑥𝐴) → (𝑥𝐴 → ∀𝑦(𝑦𝐴𝜑)))
153, 14ax-mp 7 . . 3 (𝑥𝐴 → ∀𝑦(𝑦𝐴𝜑))
16 df-ral 2364 . . 3 (∀𝑦𝐴 𝜑 ↔ ∀𝑦(𝑦𝐴𝜑))
1715, 16sylibr 132 . 2 (𝑥𝐴 → ∀𝑦𝐴 𝜑)
1817rgen 2428 1 𝑥𝐴𝑦𝐴 𝜑
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wo 664  wal 1287   = wceq 1289  wnf 1394  wcel 1438  wral 2359
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693  df-cleq 2081  df-clel 2084  df-ral 2364
This theorem is referenced by:  ordsucunielexmid  4337  onintexmid  4378  isoid  5571  issmo  6035  oawordriexmid  6213  ecopover  6370  ecopoverg  6373  1domsn  6515  unfiexmid  6608  subf  7663  negiso  8388  cnref1o  9102  ioof  9358  fzof  9520  gcdf  11057  eucalgf  11130  qredeu  11172  peano4nninf  11553
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