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Theorem rexcom13 2554
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 2553 . 2 (∃𝑥𝐴𝑦𝐵𝑧𝐶 𝜑 ↔ ∃𝑦𝐵𝑥𝐴𝑧𝐶 𝜑)
2 rexcom 2553 . . 3 (∃𝑥𝐴𝑧𝐶 𝜑 ↔ ∃𝑧𝐶𝑥𝐴 𝜑)
32rexbii 2401 . 2 (∃𝑦𝐵𝑥𝐴𝑧𝐶 𝜑 ↔ ∃𝑦𝐵𝑧𝐶𝑥𝐴 𝜑)
4 rexcom 2553 . 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 2376
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082
This theorem depends on definitions:  df-bi 116  df-tru 1302  df-nf 1405  df-sb 1704  df-cleq 2093  df-clel 2096  df-nfc 2229  df-rex 2381
This theorem is referenced by:  rexrot4  2555
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