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Theorem rgen2a 2375
 Description: Generalization rule for restricted quantification. Note that 𝑥 and 𝑦 needn't be distinct (and illustrates the use of dvelimor 1894). (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 1421 . . . . 5 𝑦 𝑧𝐴
2 eleq1 2100 . . . . 5 (𝑧 = 𝑥 → (𝑧𝐴𝑥𝐴))
31, 2dvelimor 1894 . . . 4 (∀𝑦 𝑦 = 𝑥 ∨ Ⅎ𝑦 𝑥𝐴)
4 eleq1 2100 . . . . . . . . 9 (𝑦 = 𝑥 → (𝑦𝐴𝑥𝐴))
5 rgen2a.1 . . . . . . . . . 10 ((𝑥𝐴𝑦𝐴) → 𝜑)
65ex 108 . . . . . . . . 9 (𝑥𝐴 → (𝑦𝐴𝜑))
74, 6syl6bi 152 . . . . . . . 8 (𝑦 = 𝑥 → (𝑦𝐴 → (𝑦𝐴𝜑)))
87pm2.43d 44 . . . . . . 7 (𝑦 = 𝑥 → (𝑦𝐴𝜑))
98alimi 1344 . . . . . 6 (∀𝑦 𝑦 = 𝑥 → ∀𝑦(𝑦𝐴𝜑))
109a1d 22 . . . . 5 (∀𝑦 𝑦 = 𝑥 → (𝑥𝐴 → ∀𝑦(𝑦𝐴𝜑)))
11 nfr 1411 . . . . . 6 (Ⅎ𝑦 𝑥𝐴 → (𝑥𝐴 → ∀𝑦 𝑥𝐴))
126alimi 1344 . . . . . 6 (∀𝑦 𝑥𝐴 → ∀𝑦(𝑦𝐴𝜑))
1311, 12syl6 29 . . . . 5 (Ⅎ𝑦 𝑥𝐴 → (𝑥𝐴 → ∀𝑦(𝑦𝐴𝜑)))
1410, 13jaoi 636 . . . 4 ((∀𝑦 𝑦 = 𝑥 ∨ Ⅎ𝑦 𝑥𝐴) → (𝑥𝐴 → ∀𝑦(𝑦𝐴𝜑)))
153, 14ax-mp 7 . . 3 (𝑥𝐴 → ∀𝑦(𝑦𝐴𝜑))
16 df-ral 2311 . . 3 (∀𝑦𝐴 𝜑 ↔ ∀𝑦(𝑦𝐴𝜑))
1715, 16sylibr 137 . 2 (𝑥𝐴 → ∀𝑦𝐴 𝜑)
1817rgen 2374 1 𝑥𝐴𝑦𝐴 𝜑
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ∨ wo 629  ∀wal 1241   = wceq 1243  Ⅎwnf 1349   ∈ wcel 1393  ∀wral 2306 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-cleq 2033  df-clel 2036  df-ral 2311 This theorem is referenced by:  ordsucunielexmid  4256  onintexmid  4297  isoid  5450  issmo  5903  ecopover  6204  ecopoverg  6207  subf  7213  cnref1o  8582  ioof  8840  fzof  9001
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