MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  3reeanv Structured version   Visualization version   GIF version

Theorem 3reeanv 3217
Description: Rearrange three restricted existential quantifiers. (Contributed by Jeff Madsen, 11-Jun-2010.)
Assertion
Ref Expression
3reeanv (∃𝑥𝐴𝑦𝐵𝑧𝐶 (𝜑𝜓𝜒) ↔ (∃𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜓 ∧ ∃𝑧𝐶 𝜒))
Distinct variable groups:   𝜑,𝑦,𝑧   𝜓,𝑥,𝑧   𝜒,𝑥,𝑦   𝑦,𝐴   𝑥,𝐵,𝑧   𝑥,𝐶,𝑦
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)   𝜒(𝑧)   𝐴(𝑥,𝑧)   𝐵(𝑦)   𝐶(𝑧)

Proof of Theorem 3reeanv
StepHypRef Expression
1 r19.41v 3182 . . 3 (∃𝑥𝐴 (∃𝑦𝐵 (𝜑𝜓) ∧ ∃𝑧𝐶 𝜒) ↔ (∃𝑥𝐴𝑦𝐵 (𝜑𝜓) ∧ ∃𝑧𝐶 𝜒))
2 reeanv 3216 . . . 4 (∃𝑥𝐴𝑦𝐵 (𝜑𝜓) ↔ (∃𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜓))
32anbi1i 625 . . 3 ((∃𝑥𝐴𝑦𝐵 (𝜑𝜓) ∧ ∃𝑧𝐶 𝜒) ↔ ((∃𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜓) ∧ ∃𝑧𝐶 𝜒))
41, 3bitri 275 . 2 (∃𝑥𝐴 (∃𝑦𝐵 (𝜑𝜓) ∧ ∃𝑧𝐶 𝜒) ↔ ((∃𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜓) ∧ ∃𝑧𝐶 𝜒))
5 df-3an 1090 . . . . 5 ((𝜑𝜓𝜒) ↔ ((𝜑𝜓) ∧ 𝜒))
652rexbii 3125 . . . 4 (∃𝑦𝐵𝑧𝐶 (𝜑𝜓𝜒) ↔ ∃𝑦𝐵𝑧𝐶 ((𝜑𝜓) ∧ 𝜒))
7 reeanv 3216 . . . 4 (∃𝑦𝐵𝑧𝐶 ((𝜑𝜓) ∧ 𝜒) ↔ (∃𝑦𝐵 (𝜑𝜓) ∧ ∃𝑧𝐶 𝜒))
86, 7bitri 275 . . 3 (∃𝑦𝐵𝑧𝐶 (𝜑𝜓𝜒) ↔ (∃𝑦𝐵 (𝜑𝜓) ∧ ∃𝑧𝐶 𝜒))
98rexbii 3094 . 2 (∃𝑥𝐴𝑦𝐵𝑧𝐶 (𝜑𝜓𝜒) ↔ ∃𝑥𝐴 (∃𝑦𝐵 (𝜑𝜓) ∧ ∃𝑧𝐶 𝜒))
10 df-3an 1090 . 2 ((∃𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜓 ∧ ∃𝑧𝐶 𝜒) ↔ ((∃𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜓) ∧ ∃𝑧𝐶 𝜒))
114, 9, 103bitr4i 303 1 (∃𝑥𝐴𝑦𝐵𝑧𝐶 (𝜑𝜓𝜒) ↔ (∃𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜓 ∧ ∃𝑧𝐶 𝜒))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 397  w3a 1088  wrex 3070
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914
This theorem depends on definitions:  df-bi 206  df-an 398  df-3an 1090  df-ex 1783  df-ral 3062  df-rex 3071
This theorem is referenced by:  poxp2  8076  poxp3  8083  imasmnd2  18598  imasgrp2  18867  imasring  20050  axeuclid  27954  lshpkrlem6  37623
  Copyright terms: Public domain W3C validator