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Theorem rexrot4 3082
Description: Rotate four restricted existential 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 3081 . . 3 (∃𝑦𝐵𝑧𝐶𝑤𝐷 𝜑 ↔ ∃𝑤𝐷𝑧𝐶𝑦𝐵 𝜑)
21rexbii 3023 . 2 (∃𝑥𝐴𝑦𝐵𝑧𝐶𝑤𝐷 𝜑 ↔ ∃𝑥𝐴𝑤𝐷𝑧𝐶𝑦𝐵 𝜑)
3 rexcom13 3081 . 2 (∃𝑥𝐴𝑤𝐷𝑧𝐶𝑦𝐵 𝜑 ↔ ∃𝑧𝐶𝑤𝐷𝑥𝐴𝑦𝐵 𝜑)
42, 3bitri 263 1 (∃𝑥𝐴𝑦𝐵𝑧𝐶𝑤𝐷 𝜑 ↔ ∃𝑧𝐶𝑤𝐷𝑥𝐴𝑦𝐵 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 195  wrex 2897
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902
This theorem is referenced by:  lsmspsn  18905
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