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Theorem rexcom4 2571
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 (x A yφyx A φ)
Distinct variable groups:   x,y   y,A
Allowed substitution hints:   φ(x,y)   A(x)

Proof of Theorem rexcom4
StepHypRef Expression
1 rexcom 2468 . 2 (x A y V φy V x A φ)
2 rexv 2566 . . 3 (y V φyφ)
32rexbii 2325 . 2 (x A y V φx A yφ)
4 rexv 2566 . 2 (y V x A φyx A φ)
51, 3, 43bitr3i 199 1 (x A yφyx A φ)
Colors of variables: wff set class
Syntax hints:  wb 98  wex 1378  wrex 2301  Vcvv 2551
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rex 2306  df-v 2553
This theorem is referenced by:  rexcom4a  2572  reuind  2738  iuncom4  3655  dfiun2g  3680  iunn0m  3708  iunxiun  3727  iinexgm  3899  inuni  3900  iunopab  4009  xpiundi  4341  xpiundir  4342  cnvuni  4464  dmiun  4487  elres  4589  elsnres  4590  rniun  4677  imaco  4769  coiun  4773  fun11iun  5090  abrexco  5341  imaiun  5342  fliftf  5382  rexrnmpt2  5558  oprabrexex2  5699  releldm2  5753  eroveu  6133  genpassl  6507  genpassu  6508  ltexprlemopl  6575  ltexprlemopu  6577
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