MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  r2alf Structured version   Visualization version   GIF version

Theorem r2alf 2967
Description: Double restricted universal quantification. (Contributed by Mario Carneiro, 14-Oct-2016.) Use r2allem 2966. (Revised by Wolf Lammen, 9-Jan-2020.)
Hypothesis
Ref Expression
r2alf.1 𝑦𝐴
Assertion
Ref Expression
r2alf (∀𝑥𝐴𝑦𝐵 𝜑 ↔ ∀𝑥𝑦((𝑥𝐴𝑦𝐵) → 𝜑))
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)

Proof of Theorem r2alf
StepHypRef Expression
1 r2alf.1 . . . 4 𝑦𝐴
21nfcri 2787 . . 3 𝑦 𝑥𝐴
3219.21 2113 . 2 (∀𝑦(𝑥𝐴 → (𝑦𝐵𝜑)) ↔ (𝑥𝐴 → ∀𝑦(𝑦𝐵𝜑)))
43r2allem 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  wnfc 2780  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  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946
This theorem is referenced by:  r2exf  3089  ralcomf  3125
  Copyright terms: Public domain W3C validator