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Theorem rgen2a 3369
Description: Generalization rule for restricted quantification. Note that 𝑥 and 𝑦 are not required to be disjoint. This proof illustrates the use of dvelim 2454. This theorem relies on the full set of axioms up to ax-ext 2706 and it should no longer be used. Usage of rgen2 3197 is highly encouraged. (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.) (New usage 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 2827 . . . . . 6 (𝑧 = 𝑥 → (𝑧𝐴𝑥𝐴))
21dvelimv 2455 . . . . 5 (¬ ∀𝑦 𝑦 = 𝑥 → (𝑥𝐴 → ∀𝑦 𝑥𝐴))
3 rgen2a.1 . . . . . . 7 ((𝑥𝐴𝑦𝐴) → 𝜑)
43ex 412 . . . . . 6 (𝑥𝐴 → (𝑦𝐴𝜑))
54alimi 1808 . . . . 5 (∀𝑦 𝑥𝐴 → ∀𝑦(𝑦𝐴𝜑))
62, 5syl6com 37 . . . 4 (𝑥𝐴 → (¬ ∀𝑦 𝑦 = 𝑥 → ∀𝑦(𝑦𝐴𝜑)))
7 eleq1 2827 . . . . . . 7 (𝑦 = 𝑥 → (𝑦𝐴𝑥𝐴))
87biimpd 229 . . . . . 6 (𝑦 = 𝑥 → (𝑦𝐴𝑥𝐴))
98, 4syli 39 . . . . 5 (𝑦 = 𝑥 → (𝑦𝐴𝜑))
109alimi 1808 . . . 4 (∀𝑦 𝑦 = 𝑥 → ∀𝑦(𝑦𝐴𝜑))
116, 10pm2.61d2 181 . . 3 (𝑥𝐴 → ∀𝑦(𝑦𝐴𝜑))
12 df-ral 3060 . . 3 (∀𝑦𝐴 𝜑 ↔ ∀𝑦(𝑦𝐴𝜑))
1311, 12sylibr 234 . 2 (𝑥𝐴 → ∀𝑦𝐴 𝜑)
1413rgen 3061 1 𝑥𝐴𝑦𝐴 𝜑
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wal 1535  wcel 2106  wral 3059
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-13 2375  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-nf 1781  df-cleq 2727  df-clel 2814  df-ral 3060
This theorem is referenced by: (None)
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