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Theorem r2al 2360
 Description: Double restricted universal quantification. (Contributed by NM, 19-Nov-1995.)
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 nfcv 2194 . 2 𝑦𝐴
21r2alf 2358 1 (∀𝑥𝐴𝑦𝐵 𝜑 ↔ ∀𝑥𝑦((𝑥𝐴𝑦𝐵) → 𝜑))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 101   ↔ wb 102  ∀wal 1257   ∈ wcel 1409  ∀wral 2323 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038 This theorem depends on definitions:  df-bi 114  df-nf 1366  df-sb 1662  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328 This theorem is referenced by:  r3al  2383  raliunxp  4505  codir  4741  qfto  4742  fununi  4995  dff13  5435  mpt22eqb  5638  qliftfun  6219
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