Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  ralxfr2d Unicode version

Theorem ralxfr2d 4668
 Description: Transfer universal quantification from a variable to another variable contained in expression . (Contributed by Mario Carneiro, 20-Aug-2014.)
Hypotheses
Ref Expression
ralxfr2d.1
ralxfr2d.2
ralxfr2d.3
Assertion
Ref Expression
ralxfr2d
Distinct variable groups:   ,   ,,   ,   ,   ,,   ,

Proof of Theorem ralxfr2d
StepHypRef Expression
1 ralxfr2d.1 . . . 4
2 elisset 3120 . . . 4
31, 2syl 16 . . 3
4 ralxfr2d.2 . . . . . . . 8
54biimprd 223 . . . . . . 7
6 r19.23v 2937 . . . . . . 7
75, 6sylibr 212 . . . . . 6
87r19.21bi 2826 . . . . 5
9 eleq1 2529 . . . . 5
108, 9mpbidi 216 . . . 4
1110exlimdv 1724 . . 3
123, 11mpd 15 . 2
134biimpa 484 . 2
14 ralxfr2d.3 . 2
1512, 13, 14ralxfrd 4666 1
 Colors of variables: wff setvar class Syntax hints:  ->wi 4  <->wb 184  /\wa 369  =wceq 1395  E.wex 1612  e.wcel 1818  A.wral 2807  E.wrex 2808 This theorem is referenced by:  rexxfr2d  4669  ralrn  6034  ralima  6152  cnrest2  19787  cnprest2  19791  consuba  19921  subislly  19982  trfbas2  20344  trfil2  20388  flimrest  20484  fclsrest  20525  tsmssubm  20644  metucn  21092  extoimad  37981 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-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435 This theorem depends on definitions:  df-bi 185  df-an 371  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-ral 2812  df-rex 2813  df-v 3111
 Copyright terms: Public domain W3C validator