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Theorem rgen2a 3226
Description: Generalization rule for restricted quantification. Note that 𝑥 and 𝑦 are not required to be disjoint. This proof illustrates the use of dvelim 2465. (Contributed by NM, 23-Nov-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 1-Jan-2020.) (Proof modification is discouraged.)
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 eleq1 2897 . . . . . 6 (𝑧 = 𝑥 → (𝑧𝐴𝑥𝐴))
21dvelimv 2466 . . . . 5 (¬ ∀𝑦 𝑦 = 𝑥 → (𝑥𝐴 → ∀𝑦 𝑥𝐴))
3 rgen2a.1 . . . . . . 7 ((𝑥𝐴𝑦𝐴) → 𝜑)
43ex 413 . . . . . 6 (𝑥𝐴 → (𝑦𝐴𝜑))
54alimi 1803 . . . . 5 (∀𝑦 𝑥𝐴 → ∀𝑦(𝑦𝐴𝜑))
62, 5syl6com 37 . . . 4 (𝑥𝐴 → (¬ ∀𝑦 𝑦 = 𝑥 → ∀𝑦(𝑦𝐴𝜑)))
7 eleq1 2897 . . . . . . 7 (𝑦 = 𝑥 → (𝑦𝐴𝑥𝐴))
87biimpd 230 . . . . . 6 (𝑦 = 𝑥 → (𝑦𝐴𝑥𝐴))
98, 4syli 39 . . . . 5 (𝑦 = 𝑥 → (𝑦𝐴𝜑))
109alimi 1803 . . . 4 (∀𝑦 𝑦 = 𝑥 → ∀𝑦(𝑦𝐴𝜑))
116, 10pm2.61d2 182 . . 3 (𝑥𝐴 → ∀𝑦(𝑦𝐴𝜑))
12 df-ral 3140 . . 3 (∀𝑦𝐴 𝜑 ↔ ∀𝑦(𝑦𝐴𝜑))
1311, 12sylibr 235 . 2 (𝑥𝐴 → ∀𝑦𝐴 𝜑)
1413rgen 3145 1 𝑥𝐴𝑦𝐴 𝜑
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  wal 1526  wcel 2105  wral 3135
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-13 2381  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-cleq 2811  df-clel 2890  df-ral 3140
This theorem is referenced by: (None)
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