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Theorem rgen2 2405
 Description: Generalization rule for restricted quantification. (Contributed by NM, 30-May-1999.)
Hypothesis
Ref Expression
rgen2.1 ((𝑥𝐴𝑦𝐵) → 𝜑)
Assertion
Ref Expression
rgen2 𝑥𝐴𝑦𝐵 𝜑
Distinct variable groups:   𝑥,𝑦   𝑦,𝐴
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥)   𝐵(𝑥,𝑦)

Proof of Theorem rgen2
StepHypRef Expression
1 rgen2.1 . . 3 ((𝑥𝐴𝑦𝐵) → 𝜑)
21ralrimiva 2392 . 2 (𝑥𝐴 → ∀𝑦𝐵 𝜑)
32rgen 2374 1 𝑥𝐴𝑦𝐵 𝜑
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ∈ 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-5 1336  ax-gen 1338  ax-4 1400  ax-17 1419 This theorem depends on definitions:  df-bi 110  df-nf 1350  df-ral 2311 This theorem is referenced by:  rgen3  2406  f1stres  5786  f2ndres  5787  divfnzn  8556
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