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Theorem rgen2 2422
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 2409 . 2 (𝑥𝐴 → ∀𝑦𝐵 𝜑)
32rgen 2391 1 𝑥𝐴𝑦𝐵 𝜑
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wcel 1409  wral 2323
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-gen 1354  ax-4 1416  ax-17 1435
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-ral 2328
This theorem is referenced by:  rgen3  2423  f1stres  5814  f2ndres  5815  divfnzn  8653
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