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Theorem r2al 2968
 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 (∀𝑥𝐴𝑦𝐵 𝜑 ↔ ∀𝑥𝑦((𝑥𝐴𝑦𝐵) → 𝜑))
Distinct variable groups:   𝑥,𝑦   𝑦,𝐴
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥)   𝐵(𝑥,𝑦)

Proof of Theorem r2al
StepHypRef Expression
1 19.21v 1908 . 2 (∀𝑦(𝑥𝐴 → (𝑦𝐵𝜑)) ↔ (𝑥𝐴 → ∀𝑦(𝑦𝐵𝜑)))
21r2allem 2966 1 (∀𝑥𝐴𝑦𝐵 𝜑 ↔ ∀𝑥𝑦((𝑥𝐴𝑦𝐵) → 𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383  ∀wal 1521   ∈ wcel 2030  ∀wral 2941 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879 This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1745  df-ral 2946 This theorem is referenced by:  r3al  2969  r2ex  3090  soss  5082  raliunxp  5294  codir  5551  qfto  5552  fununi  6002  dff13  6552  mpt22eqb  6811  tz7.48lem  7581  qliftfun  7875  zorn2lem4  9359  isirred2  18747  cnmpt12  21518  cnmpt22  21525  dchrelbas3  25008  cvmlift2lem12  31422  dfso2  31770  dfpo2  31771  inxpss  34223  inxpss3  34225  isdomn3  38099
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