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Theorem ralxfrd 4737
 Description: Transfer universal quantification from a variable to another variable contained in expression . (Contributed by NM, 15-Aug-2014.) (Proof shortened by Mario Carneiro, 19-Nov-2016.)
Hypotheses
Ref Expression
ralxfrd.1
ralxfrd.2
ralxfrd.3
Assertion
Ref Expression
ralxfrd
Distinct variable groups:   ,   ,,   ,   ,   ,,   ,
Allowed substitution hints:   ()   ()   ()   ()

Proof of Theorem ralxfrd
StepHypRef Expression
1 ralxfrd.1 . . . 4
2 ralxfrd.3 . . . . 5
32adantlr 696 . . . 4
41, 3rspcdv 3055 . . 3
54ralrimdva 2796 . 2
6 ralxfrd.2 . . . 4
7 r19.29 2846 . . . . 5
82biimprd 215 . . . . . . . . 9
98expimpd 587 . . . . . . . 8
109ancomsd 441 . . . . . . 7
1110ad2antrr 707 . . . . . 6
1211rexlimdva 2830 . . . . 5
137, 12syl5 30 . . . 4
146, 13mpan2d 656 . . 3
1514ralrimdva 2796 . 2
165, 15impbid 184 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wa 359   wceq 1652   wcel 1725  wral 2705  wrex 2706 This theorem is referenced by:  rexxfrd  4738  ralxfr2d  4739  ralxfr  4741  cmpfi  17471  metucnOLD  18618  metucn  18619  rlimcnp  20804  islindf4  27285  glbconN  30174  mapdordlem2  32435 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2710  df-rex 2711  df-v 2958
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