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  6321  f2ndres  6322  exmidonfinlem  7403  netap  7472  2onetap  7473  2omotaplemap  7475  mpomulf  8168  aptap  8829  divfnzn  9854  fnpfx  11257  wrd2ind  11303  1arith  12939  xpsff1o  13431  mgmidmo  13454  nmznsg  13799  isabli  13886  rhmfn  14185  cnsubmlem  14591  cnsubrglem  14593  txuni2  14979  divcnap  15288  abscncf  15308  recncf  15309  imcncf  15310  cjcncf  15311  reefiso  15500  ioocosf1o  15577  sgmf  15709  perfectlem2  15723  2lgslem1b  15817