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Theorem rexcom4 2592
Description: Commutation of restricted and unrestricted existential quantifiers. (Contributed by NM, 12-Apr-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
Assertion
Ref Expression
rexcom4 (∃𝑥𝐴𝑦𝜑 ↔ ∃𝑦𝑥𝐴 𝜑)
Distinct variable groups:   𝑥,𝑦   𝑦,𝐴
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥)

Proof of Theorem rexcom4
StepHypRef Expression
1 rexcom 2489 . 2 (∃𝑥𝐴𝑦 ∈ V 𝜑 ↔ ∃𝑦 ∈ V ∃𝑥𝐴 𝜑)
2 rexv 2587 . . 3 (∃𝑦 ∈ V 𝜑 ↔ ∃𝑦𝜑)
32rexbii 2346 . 2 (∃𝑥𝐴𝑦 ∈ V 𝜑 ↔ ∃𝑥𝐴𝑦𝜑)
4 rexv 2587 . 2 (∃𝑦 ∈ V ∃𝑥𝐴 𝜑 ↔ ∃𝑦𝑥𝐴 𝜑)
51, 3, 43bitr3i 203 1 (∃𝑥𝐴𝑦𝜑 ↔ ∃𝑦𝑥𝐴 𝜑)
Colors of variables: wff set class
Syntax hints:  wb 102  wex 1395  wrex 2322  Vcvv 2572
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 638  ax-5 1350  ax-7 1351  ax-gen 1352  ax-ie1 1396  ax-ie2 1397  ax-8 1409  ax-10 1410  ax-11 1411  ax-i12 1412  ax-bndl 1413  ax-4 1414  ax-17 1433  ax-i9 1437  ax-ial 1441  ax-i5r 1442  ax-ext 2036
This theorem depends on definitions:  df-bi 114  df-tru 1260  df-nf 1364  df-sb 1660  df-clab 2041  df-cleq 2047  df-clel 2050  df-nfc 2181  df-rex 2327  df-v 2574
This theorem is referenced by:  rexcom4a  2593  reuind  2764  iuncom4  3689  dfiun2g  3714  iunn0m  3742  iunxiun  3761  iinexgm  3933  inuni  3934  iunopab  4043  xpiundi  4423  xpiundir  4424  cnvuni  4546  dmiun  4569  elres  4671  elsnres  4672  rniun  4759  imaco  4851  coiun  4855  fun11iun  5172  abrexco  5423  imaiun  5424  fliftf  5464  rexrnmpt2  5641  oprabrexex2  5782  releldm2  5836  eroveu  6225  genpassl  6650  genpassu  6651  ltexprlemopl  6727  ltexprlemopu  6729
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