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Theorem rexprg 3776
 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 3742 . . . 4
21rexeqi 2812 . . 3
3 rexun 3443 . . 3
42, 3bitri 240 . 2
5 ralprg.1 . . . . 5
65rexsng 3766 . . . 4
76orbi1d 683 . . 3
8 ralprg.2 . . . . 5
98rexsng 3766 . . . 4
109orbi2d 682 . . 3
117, 10sylan9bb 680 . 2
124, 11syl5bb 248 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 176   wo 357   wa 358   wceq 1642   wcel 1710  wrex 2615   cun 3207  csn 3737  cpr 3738 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-rex 2620  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-un 3214  df-sn 3741  df-pr 3742 This theorem is referenced by:  rextpg  3778  rexpr  3780
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