Theorem rgen2 2618

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 2605	. 2 ⊢ (𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐵 𝜑)
3	2	rgen 2585	1 ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑

Syntax hints:   → wi 4   ∧ wa 104   ∈ wcel 2202  ∀wral 2510

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 1495  ax-gen 1497  ax-4 1558  ax-17 1574

This theorem depends on definitions:  df-bi 117  df-nf 1509  df-ral 2515

This theorem is referenced by:  rgen3  2619  invdisjrab  4082  f1stres  6322  f2ndres  6323  exmidonfinlem  7404  netap  7473  2onetap  7474  2omotaplemap  7476  mpomulf  8169  aptap  8830  divfnzn  9855  fnpfx  11262  wrd2ind  11308  1arith  12945  xpsff1o  13437  mgmidmo  13460  nmznsg  13805  isabli  13892  rhmfn  14192  cnsubmlem  14598  cnsubrglem  14600  txuni2  14986  divcnap  15295  abscncf  15315  recncf  15316  imcncf  15317  cjcncf  15318  reefiso  15507  ioocosf1o  15584  sgmf  15716  perfectlem2  15730  2lgslem1b  15824