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Theorem rexprg 3413
 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 3374 . . . 4
21rexeqi 2504 . . 3
3 rexun 3117 . . 3
42, 3bitri 173 . 2
5 ralprg.1 . . . . 5
65rexsng 3403 . . . 4
76orbi1d 704 . . 3
8 ralprg.2 . . . . 5
98rexsng 3403 . . . 4
109orbi2d 703 . . 3
117, 10sylan9bb 435 . 2
124, 11syl5bb 181 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 97   wb 98   wo 628   wceq 1242   wcel 1390  wrex 2301   cun 2909  csn 3367  cpr 3368 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-bndl 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-3an 886  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  df-sbc 2759  df-un 2916  df-sn 3373  df-pr 3374 This theorem is referenced by:  rextpg  3415  rexpr  3417
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