ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ralcom GIF version

Theorem ralcom 2523
Description: Commutation of restricted quantifiers. (Contributed by NM, 13-Oct-1999.) (Revised by Mario Carneiro, 14-Oct-2016.)
Assertion
Ref Expression
ralcom (∀𝑥𝐴𝑦𝐵 𝜑 ↔ ∀𝑦𝐵𝑥𝐴 𝜑)
Distinct variable groups:   𝑥,𝑦   𝑥,𝐵   𝑦,𝐴
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥)   𝐵(𝑦)

Proof of Theorem ralcom
StepHypRef Expression
1 nfcv 2223 . 2 𝑦𝐴
2 nfcv 2223 . 2 𝑥𝐵
31, 2ralcomf 2521 1 (∀𝑥𝐴𝑦𝐵 𝜑 ↔ ∀𝑦𝐵𝑥𝐴 𝜑)
Colors of variables: wff set class
Syntax hints:  wb 103  wral 2353
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1688  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358
This theorem is referenced by:  ralcom4  2632  ssint  3678  issod  4109  reusv3  4245  cnvpom  4925  cnvsom  4926  fununi  5033  isocnv2  5529  dfsmo2  5982  rexfiuz  10247
  Copyright terms: Public domain W3C validator