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Theorem 3reeanv 3367
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 3345 . . 3 (∃𝑥𝐴 (∃𝑦𝐵 (𝜑𝜓) ∧ ∃𝑧𝐶 𝜒) ↔ (∃𝑥𝐴𝑦𝐵 (𝜑𝜓) ∧ ∃𝑧𝐶 𝜒))
2 reeanv 3366 . . . 4 (∃𝑥𝐴𝑦𝐵 (𝜑𝜓) ↔ (∃𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜓))
32anbi1i 625 . . 3 ((∃𝑥𝐴𝑦𝐵 (𝜑𝜓) ∧ ∃𝑧𝐶 𝜒) ↔ ((∃𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜓) ∧ ∃𝑧𝐶 𝜒))
41, 3bitri 277 . 2 (∃𝑥𝐴 (∃𝑦𝐵 (𝜑𝜓) ∧ ∃𝑧𝐶 𝜒) ↔ ((∃𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜓) ∧ ∃𝑧𝐶 𝜒))
5 df-3an 1083 . . . . 5 ((𝜑𝜓𝜒) ↔ ((𝜑𝜓) ∧ 𝜒))
652rexbii 3246 . . . 4 (∃𝑦𝐵𝑧𝐶 (𝜑𝜓𝜒) ↔ ∃𝑦𝐵𝑧𝐶 ((𝜑𝜓) ∧ 𝜒))
7 reeanv 3366 . . . 4 (∃𝑦𝐵𝑧𝐶 ((𝜑𝜓) ∧ 𝜒) ↔ (∃𝑦𝐵 (𝜑𝜓) ∧ ∃𝑧𝐶 𝜒))
86, 7bitri 277 . . 3 (∃𝑦𝐵𝑧𝐶 (𝜑𝜓𝜒) ↔ (∃𝑦𝐵 (𝜑𝜓) ∧ ∃𝑧𝐶 𝜒))
98rexbii 3245 . 2 (∃𝑥𝐴𝑦𝐵𝑧𝐶 (𝜑𝜓𝜒) ↔ ∃𝑥𝐴 (∃𝑦𝐵 (𝜑𝜓) ∧ ∃𝑧𝐶 𝜒))
10 df-3an 1083 . 2 ((∃𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜓 ∧ ∃𝑧𝐶 𝜒) ↔ ((∃𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜓) ∧ ∃𝑧𝐶 𝜒))
114, 9, 103bitr4i 305 1 (∃𝑥𝐴𝑦𝐵𝑧𝐶 (𝜑𝜓𝜒) ↔ (∃𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜓 ∧ ∃𝑧𝐶 𝜒))
Colors of variables: wff setvar class
Syntax hints:  wb 208  wa 398  w3a 1081  wrex 3137
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904
This theorem depends on definitions:  df-bi 209  df-an 399  df-3an 1083  df-ex 1774  df-ral 3141  df-rex 3142
This theorem is referenced by:  imasmnd2  17940  imasgrp2  18206  imasring  19361  axeuclid  26741  lshpkrlem6  36243
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