Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  rexrot4 GIF version

Theorem rexrot4 2601
 Description: Rotate existential restricted quantifiers twice. (Contributed by NM, 8-Apr-2015.)
Assertion
Ref Expression
rexrot4 (∃𝑥𝐴𝑦𝐵𝑧𝐶𝑤𝐷 𝜑 ↔ ∃𝑧𝐶𝑤𝐷𝑥𝐴𝑦𝐵 𝜑)
Distinct variable groups:   𝑧,𝑤,𝐴   𝑤,𝐵,𝑧   𝑥,𝑤,𝑦,𝐶   𝑥,𝑧,𝐷,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑤)   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)   𝐶(𝑧)   𝐷(𝑤)

Proof of Theorem rexrot4
StepHypRef Expression
1 rexcom13 2600 . . 3 (∃𝑦𝐵𝑧𝐶𝑤𝐷 𝜑 ↔ ∃𝑤𝐷𝑧𝐶𝑦𝐵 𝜑)
21rexbii 2446 . 2 (∃𝑥𝐴𝑦𝐵𝑧𝐶𝑤𝐷 𝜑 ↔ ∃𝑥𝐴𝑤𝐷𝑧𝐶𝑦𝐵 𝜑)
3 rexcom13 2600 . 2 (∃𝑥𝐴𝑤𝐷𝑧𝐶𝑦𝐵 𝜑 ↔ ∃𝑧𝐶𝑤𝐷𝑥𝐴𝑦𝐵 𝜑)
42, 3bitri 183 1 (∃𝑥𝐴𝑦𝐵𝑧𝐶𝑤𝐷 𝜑 ↔ ∃𝑧𝐶𝑤𝐷𝑥𝐴𝑦𝐵 𝜑)
 Colors of variables: wff set class Syntax hints:   ↔ wb 104  ∃wrex 2418 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122 This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-cleq 2133  df-clel 2136  df-nfc 2271  df-rex 2423 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator