Theorem r2al 3119

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 2035	. 2 ⊢ (∀𝑦(𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐵 → 𝜑)) ↔ (𝑥 ∈ 𝐴 → ∀𝑦(𝑦 ∈ 𝐵 → 𝜑)))
2	1	r2allem 3117	1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝜑))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 385  ∀wal 1651   ∈ wcel 2157  ∀wral 3088

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006

This theorem depends on definitions:  df-bi 199  df-an 386  df-ex 1876  df-ral 3093

This theorem is referenced by:  r3al  3120  r2ex  3241  soss  5250  raliunxp  5464  codir  5733  qfto  5734  fununi  6174  dff13  6739  mpt22eqb  7002  tz7.48lem  7774  qliftfun  8069  zorn2lem4  9608  isirred2  19014  cnmpt12  21796  cnmpt22  21803  dchrelbas3  25312  cvmlift2lem12  31806  dfso2  32151  dfpo2  32152  inxpss  34570  inxpss3  34572  isdomn3  38556