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Theorem rexrot4 3101
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 3099 . . 3 (∃𝑦𝐵𝑧𝐶𝑤𝐷 𝜑 ↔ ∃𝑤𝐷𝑧𝐶𝑦𝐵 𝜑)
21rexbii 3039 . 2 (∃𝑥𝐴𝑦𝐵𝑧𝐶𝑤𝐷 𝜑 ↔ ∃𝑥𝐴𝑤𝐷𝑧𝐶𝑦𝐵 𝜑)
3 rexcom13 3099 . 2 (∃𝑥𝐴𝑤𝐷𝑧𝐶𝑦𝐵 𝜑 ↔ ∃𝑧𝐶𝑤𝐷𝑥𝐴𝑦𝐵 𝜑)
42, 3bitri 264 1 (∃𝑥𝐴𝑦𝐵𝑧𝐶𝑤𝐷 𝜑 ↔ ∃𝑧𝐶𝑤𝐷𝑥𝐴𝑦𝐵 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 196  wrex 2912
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ral 2916  df-rex 2917
This theorem is referenced by:  lsmspsn  19078
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