Theorem rgen2 2636

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 2623	. 2 ⊢ (𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐵 𝜑)
3	2	rgen 2603	1 ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ∈ wcel 2209  ∀wral 2528

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-gen 1502  ax-4 1563  ax-17 1579

This proof depends on definitions:  df-bi 117  df-nf 1514  df-ral 2533

This theorem is used by:  rgen3  2637  invdisjrab  4124  f1stres  6393  f2ndres  6394  exmidonfinlem  7546  netap  7621  2onetap  7622  2omotaplemap  7624  mpomulf  8317  aptap  8981  zfidc  9728  divfnzn  10031  fnpfx  11464  wrd2ind  11510  1arith  13168  ballotfilem2  13279  xpsff1o  13721  mgmidmo  13743  nmznsg  14067  isabli  14154  rhmfn  14530  cnsubmlem  14966  cnsubrglem  14968  txuni2  15409  divcnap  15718  abscncf  15738  recncf  15739  imcncf  15740  cjcncf  15741  reefiso  15930  ioocosf1o  16008  sgmf  16177  perfectlem2  16222  2lgslem1b  16330