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Theorem rgen2 2540
 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 2527 . 2 (𝑥𝐴 → ∀𝑦𝐵 𝜑)
32rgen 2507 1 𝑥𝐴𝑦𝐵 𝜑
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ∈ wcel 2125  ∀wral 2432 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1424  ax-gen 1426  ax-4 1487  ax-17 1503 This theorem depends on definitions:  df-bi 116  df-nf 1438  df-ral 2437 This theorem is referenced by:  rgen3  2541  f1stres  6097  f2ndres  6098  exmidonfinlem  7107  divfnzn  9508  txuni2  12595  divcnap  12894  abscncf  12911  recncf  12912  imcncf  12913  cjcncf  12914  reefiso  13037  ioocosf1o  13114
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