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Theorem rgen2 2495
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 2482 . 2 (𝑥𝐴 → ∀𝑦𝐵 𝜑)
32rgen 2462 1 𝑥𝐴𝑦𝐵 𝜑
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wcel 1465  wral 2393
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 1408  ax-gen 1410  ax-4 1472  ax-17 1491
This theorem depends on definitions:  df-bi 116  df-nf 1422  df-ral 2398
This theorem is referenced by:  rgen3  2496  f1stres  6025  f2ndres  6026  exmidonfinlem  7017  divfnzn  9381  txuni2  12352  divcnap  12651  abscncf  12668  recncf  12669  imcncf  12670  cjcncf  12671
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