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

Theorem ralxp 5149
 Description: Universal quantification restricted to a Cartesian product is equivalent to a double restricted quantification. The hypothesis specifies an implicit substitution. (Contributed by NM, 7-Feb-2004.) (Revised by Mario Carneiro, 29-Dec-2014.)
Hypothesis
Ref Expression
ralxp.1
Assertion
Ref Expression
ralxp
Distinct variable groups:   ,,,   ,,   ,,   ,   ,

Proof of Theorem ralxp
StepHypRef Expression
1 iunxpconst 5061 . . 3
21raleqi 3058 . 2
3 ralxp.1 . . 3
43raliunxp 5147 . 2
52, 4bitr3i 251 1
 Colors of variables: wff setvar class Syntax hints:  ->wi 4  <->wb 184  =wceq 1395  A.wral 2807  {csn 4029  <.cop 4035  U_ciun 4330  X.cxp 5002 This theorem is referenced by:  ralxpf  5154  issref  5385  ffnov  6406  eqfnov  6408  funimassov  6452  f1stres  6822  f2ndres  6823  ecopover  7434  xpf1o  7699  xpwdomg  8032  rankxplim  8318  imasaddfnlem  14925  imasvscafn  14934  comfeq  15101  isssc  15189  isfuncd  15234  cofucl  15257  funcres2b  15266  evlfcl  15491  uncfcurf  15508  yonedalem3  15549  yonedainv  15550  efgval2  16742  txbas  20068  hausdiag  20146  tx1stc  20151  txkgen  20153  xkococn  20161  cnmpt21  20172  xkoinjcn  20188  tmdcn2  20588  clssubg  20607  qustgplem  20619  txmetcnp  21050  txmetcn  21051  qtopbaslem  21265  bndth  21458  cxpcn3  23122  dvdsmulf1o  23470  fsumdvdsmul  23471  xrofsup  27582  txpcon  28677  cvmlift2lem1  28747  cvmlift2lem12  28759  mclsax  28929  f1opr  30215  ismtyhmeolem  30300  ffnaov  32284  ovn0ssdmfun  32455  plusfreseq  32460  dih1dimatlem  37056 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3435  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-iun 4332  df-opab 4511  df-xp 5010  df-rel 5011
 Copyright terms: Public domain W3C validator