Theorem r2al 2561

Description: Double restricted universal quantification. (Contributed by NM, 19-Nov-1995.)

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		nfcv 2384	. 2 ⊢ Ⅎ𝑦𝐴
2	1	r2alf 2559	1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝜑))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  ∀wal 1396   ∈ wcel 2203  ∀wral 2520

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-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1812  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525

This theorem is referenced by:  r3al  2586  raliunxp  4896  codir  5151  qfto  5152  fununi  5424  dff13  5941  mpo2eqb  6163  qliftfun  6851  cnmpt12  15152  cnmpt22  15159