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Theorem ralcom4 3197
 Description: Commutation of restricted and unrestricted universal quantifiers. (Contributed by NM, 26-Mar-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
Assertion
Ref Expression
ralcom4 (∀𝑥𝐴𝑦𝜑 ↔ ∀𝑦𝑥𝐴 𝜑)
Distinct variable groups:   𝑥,𝑦   𝑦,𝐴
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥)

Proof of Theorem ralcom4
StepHypRef Expression
1 ralcom 3079 . 2 (∀𝑥𝐴𝑦 ∈ V 𝜑 ↔ ∀𝑦 ∈ V ∀𝑥𝐴 𝜑)
2 ralv 3192 . . 3 (∀𝑦 ∈ V 𝜑 ↔ ∀𝑦𝜑)
32ralbii 2963 . 2 (∀𝑥𝐴𝑦 ∈ V 𝜑 ↔ ∀𝑥𝐴𝑦𝜑)
4 ralv 3192 . 2 (∀𝑦 ∈ V ∀𝑥𝐴 𝜑 ↔ ∀𝑦𝑥𝐴 𝜑)
51, 3, 43bitr3i 289 1 (∀𝑥𝐴𝑦𝜑 ↔ ∀𝑦𝑥𝐴 𝜑)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195  ∀wal 1473  ∀wral 2896  Vcvv 3173 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-v 3175 This theorem is referenced by:  ralxpxfr2d  3298  uniiunlem  3653  iunss  4497  disjor  4567  trint  4696  reliun  5162  funimass4  6157  ralrnmpt2  6673  findcard3  8088  kmlem12  8866  fimaxre3  10849  vdwmc2  15521  ramtlecl  15542  iunocv  19844  1stccn  21076  itg2leub  23307  mptelee  25575  nmoubi  27011  nmopub  28151  nmfnleub  28168  moel  28707  disjorf  28774  funcnv5mpt  28852  untuni  30840  heibor1lem  32778  pmapglbx  34073  ss2iundf  36970  iunssf  38290  setrec1lem2  42234
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