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Theorem rexcom4 2633
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 2524 . 2 (∃𝑥𝐴𝑦 ∈ V 𝜑 ↔ ∃𝑦 ∈ V ∃𝑥𝐴 𝜑)
2 rexv 2628 . . 3 (∃𝑦 ∈ V 𝜑 ↔ ∃𝑦𝜑)
32rexbii 2379 . 2 (∃𝑥𝐴𝑦 ∈ V 𝜑 ↔ ∃𝑥𝐴𝑦𝜑)
4 rexv 2628 . 2 (∃𝑦 ∈ V ∃𝑥𝐴 𝜑 ↔ ∃𝑦𝑥𝐴 𝜑)
51, 3, 43bitr3i 208 1 (∃𝑥𝐴𝑦𝜑 ↔ ∃𝑦𝑥𝐴 𝜑)
Colors of variables: wff set class
Syntax hints:  wb 103  wex 1422  wrex 2354  Vcvv 2612
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-rex 2359  df-v 2614
This theorem is referenced by:  rexcom4a  2634  reuind  2806  iuncom4  3711  dfiun2g  3736  iunn0m  3764  iunxiun  3783  iinexgm  3955  inuni  3956  iunopab  4071  xpiundi  4452  xpiundir  4453  cnvuni  4578  dmiun  4601  elres  4703  elsnres  4704  rniun  4794  imaco  4888  coiun  4892  fun11iun  5220  abrexco  5476  imaiun  5477  fliftf  5516  rexrnmpt2  5693  oprabrexex2  5834  releldm2  5888  eroveu  6311  genpassl  6984  genpassu  6985  ltexprlemopl  7061  ltexprlemopu  7063
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