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Theorem ralprg 3775
 Description: Convert a quantification over a pair to a conjunction. (Contributed by NM, 17-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.)
Hypotheses
Ref Expression
ralprg.1
ralprg.2
Assertion
Ref Expression
ralprg
Distinct variable groups:   ,   ,   ,   ,
Allowed substitution hints:   ()   ()   ()

Proof of Theorem ralprg
StepHypRef Expression
1 df-pr 3742 . . . 4
21raleqi 2811 . . 3
3 ralunb 3444 . . 3
42, 3bitri 240 . 2
5 ralprg.1 . . . 4
65ralsng 3765 . . 3
7 ralprg.2 . . . 4
87ralsng 3765 . . 3
96, 8bi2anan9 843 . 2
104, 9syl5bb 248 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 176   wa 358   wceq 1642   wcel 1710  wral 2614   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-ral 2619  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:  raltpg  3777  ralpr  3779  iinxprg  4043
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