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Theorem rexcom13 2573
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 2572 . 2 (∃𝑥𝐴𝑦𝐵𝑧𝐶 𝜑 ↔ ∃𝑦𝐵𝑥𝐴𝑧𝐶 𝜑)
2 rexcom 2572 . . 3 (∃𝑥𝐴𝑧𝐶 𝜑 ↔ ∃𝑧𝐶𝑥𝐴 𝜑)
32rexbii 2419 . 2 (∃𝑦𝐵𝑥𝐴𝑧𝐶 𝜑 ↔ ∃𝑦𝐵𝑧𝐶𝑥𝐴 𝜑)
4 rexcom 2572 . 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 2394
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-cleq 2110  df-clel 2113  df-nfc 2247  df-rex 2399
This theorem is referenced by:  rexrot4  2574
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