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Theorem rexprg 3452
 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 3413 . . . 4
21rexeqi 2555 . . 3
3 rexun 3153 . . 3
42, 3bitri 182 . 2
5 ralprg.1 . . . . 5
65rexsng 3442 . . . 4
76orbi1d 738 . . 3
8 ralprg.2 . . . . 5
98rexsng 3442 . . . 4
109orbi2d 737 . . 3
117, 10sylan9bb 450 . 2
124, 11syl5bb 190 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 102   wb 103   wo 662   wceq 1285   wcel 1434  wrex 2350   cun 2972  csn 3406  cpr 3407 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 2064 This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-rex 2355  df-v 2604  df-sbc 2817  df-un 2978  df-sn 3412  df-pr 3413 This theorem is referenced by:  rextpg  3454  rexpr  3456  minmax  10250
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