Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  rexcom13 GIF version

Theorem rexcom13 2601
 Description: Swap 1st and 3rd restricted existential quantifiers. (Contributed by NM, 8-Apr-2015.)
Assertion
Ref Expression
rexcom13 (∃𝑥𝐴𝑦𝐵𝑧𝐶 𝜑 ↔ ∃𝑧𝐶𝑦𝐵𝑥𝐴 𝜑)
Distinct variable groups:   𝑦,𝑧,𝐴   𝑥,𝑧,𝐵   𝑥,𝑦,𝐶
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝐴(𝑥)   𝐵(𝑦)   𝐶(𝑧)

Proof of Theorem rexcom13
StepHypRef Expression
1 rexcom 2600 . 2 (∃𝑥𝐴𝑦𝐵𝑧𝐶 𝜑 ↔ ∃𝑦𝐵𝑥𝐴𝑧𝐶 𝜑)
2 rexcom 2600 . . 3 (∃𝑥𝐴𝑧𝐶 𝜑 ↔ ∃𝑧𝐶𝑥𝐴 𝜑)
32rexbii 2447 . 2 (∃𝑦𝐵𝑥𝐴𝑧𝐶 𝜑 ↔ ∃𝑦𝐵𝑧𝐶𝑥𝐴 𝜑)
4 rexcom 2600 . 2 (∃𝑦𝐵𝑧𝐶𝑥𝐴 𝜑 ↔ ∃𝑧𝐶𝑦𝐵𝑥𝐴 𝜑)
51, 3, 43bitri 205 1 (∃𝑥𝐴𝑦𝐵𝑧𝐶 𝜑 ↔ ∃𝑧𝐶𝑦𝐵𝑥𝐴 𝜑)
 Colors of variables: wff set class Syntax hints:   ↔ wb 104  ∃wrex 2419 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 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-ext 2123 This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1732  df-cleq 2134  df-clel 2137  df-nfc 2272  df-rex 2424 This theorem is referenced by:  rexrot4  2602
 Copyright terms: Public domain W3C validator