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  6244  f2ndres  6245  exmidonfinlem  7300  netap  7365  2onetap  7366  2omotaplemap  7368  mpomulf  8061  aptap  8722  divfnzn  9741  1arith  12632  xpsff1o  13123  mgmidmo  13146  nmznsg  13491  isabli  13578  rhmfn  13876  cnsubmlem  14282  cnsubrglem  14284  txuni2  14670  divcnap  14979  abscncf  14999  recncf  15000  imcncf  15001  cjcncf  15002  reefiso  15191  ioocosf1o  15268  sgmf  15400  perfectlem2  15414  2lgslem1b  15508