Theorem rgen2 2591

Description: Generalization rule for restricted quantification. (Contributed by NM, 30-May-1999.)

Hypothesis

Ref	Expression
rgen2.1	⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝜑)

Assertion

Ref	Expression
rgen2	⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑

Distinct variable groups:   𝑥,𝑦   𝑦,𝐴

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥)   𝐵(𝑥,𝑦)

Proof of Theorem rgen2

Step	Hyp	Ref	Expression
1		rgen2.1	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝜑)
2	1	ralrimiva 2578	. 2 ⊢ (𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐵 𝜑)
3	2	rgen 2558	1 ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑

Syntax hints:   → wi 4   ∧ wa 104   ∈ wcel 2175  ∀wral 2483

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1469  ax-gen 1471  ax-4 1532  ax-17 1548

This theorem depends on definitions:  df-bi 117  df-nf 1483  df-ral 2488

This theorem is referenced by:  rgen3  2592  f1stres  6235  f2ndres  6236  exmidonfinlem  7283  netap  7348  2onetap  7349  2omotaplemap  7351  mpomulf  8044  aptap  8705  divfnzn  9724  1arith  12609  xpsff1o  13099  mgmidmo  13122  nmznsg  13467  isabli  13554  rhmfn  13852  cnsubmlem  14258  cnsubrglem  14260  txuni2  14646  divcnap  14955  abscncf  14975  recncf  14976  imcncf  14977  cjcncf  14978  reefiso  15167  ioocosf1o  15244  sgmf  15376  perfectlem2  15390  2lgslem1b  15484