Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  rexprg Structured version   Unicode version

Theorem rexprg 4044
 Description: Convert a quantification over a pair to a disjunction. (Contributed by NM, 17-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.)
Hypotheses
Ref Expression
ralprg.1
ralprg.2
Assertion
Ref Expression
rexprg
Distinct variable groups:   ,   ,   ,   ,
Allowed substitution hints:   ()   ()   ()

Proof of Theorem rexprg
StepHypRef Expression
1 df-pr 3996 . . . 4
21rexeqi 3028 . . 3
3 rexun 3643 . . 3
42, 3bitri 252 . 2
5 ralprg.1 . . . . 5
65rexsng 4029 . . . 4
76orbi1d 707 . . 3
8 ralprg.2 . . . . 5
98rexsng 4029 . . . 4
109orbi2d 706 . . 3
117, 10sylan9bb 704 . 2
124, 11syl5bb 260 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 187   wo 369   wa 370   wceq 1437   wcel 1867  wrex 2774   cun 3431  csn 3993  cpr 3995 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398 This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-rex 2779  df-v 3080  df-sbc 3297  df-un 3438  df-sn 3994  df-pr 3996 This theorem is referenced by:  rextpg  4046  rexpr  4048  fr2nr  4823  sgrp2nmndlem5  16615  nb3graprlem2  25066  frgra2v  25613  3vfriswmgralem  25618  brfvrcld  35970  rnmptpr  37114  ldepspr  39083  zlmodzxzldeplem4  39113
 Copyright terms: Public domain W3C validator