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Theorem rgen2a 2422
Description: Generalization rule for restricted quantification. Note that 𝑥 and 𝑦 needn't be distinct (and illustrates the use of dvelimor 1937). (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 1462 . . . . 5 𝑦 𝑧𝐴
2 eleq1 2145 . . . . 5 (𝑧 = 𝑥 → (𝑧𝐴𝑥𝐴))
31, 2dvelimor 1937 . . . 4 (∀𝑦 𝑦 = 𝑥 ∨ Ⅎ𝑦 𝑥𝐴)
4 eleq1 2145 . . . . . . . . 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 1385 . . . . . 6 (∀𝑦 𝑦 = 𝑥 → ∀𝑦(𝑦𝐴𝜑))
109a1d 22 . . . . 5 (∀𝑦 𝑦 = 𝑥 → (𝑥𝐴 → ∀𝑦(𝑦𝐴𝜑)))
11 nfr 1452 . . . . . 6 (Ⅎ𝑦 𝑥𝐴 → (𝑥𝐴 → ∀𝑦 𝑥𝐴))
126alimi 1385 . . . . . 6 (∀𝑦 𝑥𝐴 → ∀𝑦(𝑦𝐴𝜑))
1311, 12syl6 33 . . . . 5 (Ⅎ𝑦 𝑥𝐴 → (𝑥𝐴 → ∀𝑦(𝑦𝐴𝜑)))
1410, 13jaoi 669 . . . 4 ((∀𝑦 𝑦 = 𝑥 ∨ Ⅎ𝑦 𝑥𝐴) → (𝑥𝐴 → ∀𝑦(𝑦𝐴𝜑)))
153, 14ax-mp 7 . . 3 (𝑥𝐴 → ∀𝑦(𝑦𝐴𝜑))
16 df-ral 2358 . . 3 (∀𝑦𝐴 𝜑 ↔ ∀𝑦(𝑦𝐴𝜑))
1715, 16sylibr 132 . 2 (𝑥𝐴 → ∀𝑦𝐴 𝜑)
1817rgen 2421 1 𝑥𝐴𝑦𝐴 𝜑
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wo 662  wal 1283   = wceq 1285  wnf 1390  wcel 1434  wral 2353
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1688  df-cleq 2076  df-clel 2079  df-ral 2358
This theorem is referenced by:  ordsucunielexmid  4302  onintexmid  4343  isoid  5501  issmo  5957  oawordriexmid  6134  ecopover  6291  ecopoverg  6294  1domsn  6384  unfiexmid  6462  subf  7429  negiso  8152  cnref1o  8866  ioof  9122  fzof  9283  gcdf  10571  eucalgf  10644  qredeu  10686
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