Theorem r2al 3187

Description: Double restricted universal quantification. (Contributed by NM, 19-Nov-1995.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 9-Jan-2020.)

Assertion

Ref	Expression
r2al	⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑥∀𝑦((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 r2al

Step	Hyp	Ref	Expression
1		19.21v 1942	. 2 ⊢ (∀𝑦(𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐵 → 𝜑)) ↔ (𝑥 ∈ 𝐴 → ∀𝑦(𝑦 ∈ 𝐵 → 𝜑)))
2	1	r2allem 3135	1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝜑))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396  ∀wal 1539   ∈ wcel 2106  ∀wral 3060

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913

This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1782  df-ral 3061

This theorem is referenced by:  r2ex  3188  r3al  3189  ralcom  3270  nfra2w  3280  moel  3373  raliunxp  5800  codir  6079  qfto  6080  dfpo2  6253  fununi  6581  dff13  7207  mpo2eqb  7493  frpoins3xpg  8077  xpord2indlem  8084  tz7.48lem  8392  qliftfun  8748  zorn2lem4  10444  isirred2  20146  cnmpt12  23055  cnmpt22  23062  dchrelbas3  26623  cvmlift2lem12  33995  dfso2  34414  r2alan  36780  inxpss  36845  inxpss3  36848  isdomn3  41589  iscnrm3lem2  47087  joindm2  47121  meetdm2  47123