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Theorem rexcom4 2681
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 2570 . 2 (∃𝑥𝐴𝑦 ∈ V 𝜑 ↔ ∃𝑦 ∈ V ∃𝑥𝐴 𝜑)
2 rexv 2676 . . 3 (∃𝑦 ∈ V 𝜑 ↔ ∃𝑦𝜑)
32rexbii 2417 . 2 (∃𝑥𝐴𝑦 ∈ V 𝜑 ↔ ∃𝑥𝐴𝑦𝜑)
4 rexv 2676 . 2 (∃𝑦 ∈ V ∃𝑥𝐴 𝜑 ↔ ∃𝑦𝑥𝐴 𝜑)
51, 3, 43bitr3i 209 1 (∃𝑥𝐴𝑦𝜑 ↔ ∃𝑦𝑥𝐴 𝜑)
Colors of variables: wff set class
Syntax hints:  wb 104  wex 1451  wrex 2392  Vcvv 2658
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-rex 2397  df-v 2660
This theorem is referenced by:  rexcom4a  2682  reuind  2860  iuncom4  3788  dfiun2g  3813  iunn0m  3841  iunxiun  3862  iinexgm  4047  inuni  4048  iunopab  4171  xpiundi  4565  xpiundir  4566  cnvuni  4693  dmiun  4716  elres  4823  elsnres  4824  rniun  4917  imaco  5012  coiun  5016  fun11iun  5354  abrexco  5626  imaiun  5627  fliftf  5666  rexrnmpo  5852  oprabrexex2  5994  releldm2  6049  eroveu  6486  genpassl  7296  genpassu  7297  ltexprlemopl  7373  ltexprlemopu  7375  ntreq0  12196  metrest  12570
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