Theorem rgen2 2616

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 2603	. 2 ⊢ (𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐵 𝜑)
3	2	rgen 2583	1 ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑

Syntax hints:   → wi 4   ∧ wa 104   ∈ wcel 2200  ∀wral 2508

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 1493  ax-gen 1495  ax-4 1556  ax-17 1572

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-ral 2513

This theorem is referenced by:  rgen3  2617  invdisjrab  4078  f1stres  6315  f2ndres  6316  exmidonfinlem  7392  netap  7461  2onetap  7462  2omotaplemap  7464  mpomulf  8157  aptap  8818  divfnzn  9843  fnpfx  11245  wrd2ind  11291  1arith  12927  xpsff1o  13419  mgmidmo  13442  nmznsg  13787  isabli  13874  rhmfn  14173  cnsubmlem  14579  cnsubrglem  14581  txuni2  14967  divcnap  15276  abscncf  15296  recncf  15297  imcncf  15298  cjcncf  15299  reefiso  15488  ioocosf1o  15565  sgmf  15697  perfectlem2  15711  2lgslem1b  15805