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Theorem rgen2a 2460
Description: Generalization rule for restricted quantification. Note that 𝑥 and 𝑦 needn't be distinct (and illustrates the use of dvelimor 1969). (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 1491 . . . . 5 𝑦 𝑧𝐴
2 eleq1 2177 . . . . 5 (𝑧 = 𝑥 → (𝑧𝐴𝑥𝐴))
31, 2dvelimor 1969 . . . 4 (∀𝑦 𝑦 = 𝑥 ∨ Ⅎ𝑦 𝑥𝐴)
4 eleq1 2177 . . . . . . . . 9 (𝑦 = 𝑥 → (𝑦𝐴𝑥𝐴))
5 rgen2a.1 . . . . . . . . . 10 ((𝑥𝐴𝑦𝐴) → 𝜑)
65ex 114 . . . . . . . . 9 (𝑥𝐴 → (𝑦𝐴𝜑))
74, 6syl6bi 162 . . . . . . . 8 (𝑦 = 𝑥 → (𝑦𝐴 → (𝑦𝐴𝜑)))
87pm2.43d 50 . . . . . . 7 (𝑦 = 𝑥 → (𝑦𝐴𝜑))
98alimi 1414 . . . . . 6 (∀𝑦 𝑦 = 𝑥 → ∀𝑦(𝑦𝐴𝜑))
109a1d 22 . . . . 5 (∀𝑦 𝑦 = 𝑥 → (𝑥𝐴 → ∀𝑦(𝑦𝐴𝜑)))
11 nfr 1481 . . . . . 6 (Ⅎ𝑦 𝑥𝐴 → (𝑥𝐴 → ∀𝑦 𝑥𝐴))
126alimi 1414 . . . . . 6 (∀𝑦 𝑥𝐴 → ∀𝑦(𝑦𝐴𝜑))
1311, 12syl6 33 . . . . 5 (Ⅎ𝑦 𝑥𝐴 → (𝑥𝐴 → ∀𝑦(𝑦𝐴𝜑)))
1410, 13jaoi 688 . . . 4 ((∀𝑦 𝑦 = 𝑥 ∨ Ⅎ𝑦 𝑥𝐴) → (𝑥𝐴 → ∀𝑦(𝑦𝐴𝜑)))
153, 14ax-mp 7 . . 3 (𝑥𝐴 → ∀𝑦(𝑦𝐴𝜑))
16 df-ral 2395 . . 3 (∀𝑦𝐴 𝜑 ↔ ∀𝑦(𝑦𝐴𝜑))
1715, 16sylibr 133 . 2 (𝑥𝐴 → ∀𝑦𝐴 𝜑)
1817rgen 2459 1 𝑥𝐴𝑦𝐴 𝜑
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wo 680  wal 1312   = wceq 1314  wnf 1419  wcel 1463  wral 2390
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-nf 1420  df-sb 1719  df-cleq 2108  df-clel 2111  df-ral 2395
This theorem is referenced by:  ordsucunielexmid  4406  onintexmid  4447  isoid  5665  issmo  6139  oawordriexmid  6320  ecopover  6481  ecopoverg  6484  1domsn  6666  unfiexmid  6759  axaddf  7603  axmulf  7604  subf  7887  negiso  8623  cnref1o  9342  xaddf  9520  ioof  9647  fzof  9814  xrnegiso  10923  reeff1  11258  gcdf  11509  eucalgf  11582  qredeu  11624  qnnen  11789  strsetsid  11835  ismeti  12335  qtopbasss  12510  tgqioo  12533  peano4nninf  12892
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