Theorem rgen2 2619

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 2606	. 2 ⊢ (𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐵 𝜑)
3	2	rgen 2586	1 ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑

Syntax hints:   → wi 4   ∧ wa 104   ∈ wcel 2202  ∀wral 2511

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 1496  ax-gen 1498  ax-4 1559  ax-17 1575

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-ral 2516

This theorem is referenced by:  rgen3  2620  invdisjrab  4087  f1stres  6331  f2ndres  6332  exmidonfinlem  7447  netap  7516  2onetap  7517  2omotaplemap  7519  mpomulf  8212  aptap  8872  divfnzn  9899  fnpfx  11307  wrd2ind  11353  1arith  13003  xpsff1o  13495  mgmidmo  13518  nmznsg  13863  isabli  13950  rhmfn  14250  cnsubmlem  14657  cnsubrglem  14659  txuni2  15050  divcnap  15359  abscncf  15379  recncf  15380  imcncf  15381  cjcncf  15382  reefiso  15571  ioocosf1o  15648  sgmf  15783  perfectlem2  15797  2lgslem1b  15891