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Theorem ralxfrd 4636
 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 719 . . . 4
41, 3rspcdv 3191 . . 3
54ralrimdva 2850 . 2
6 ralxfrd.2 . . . 4
7 r19.29 2970 . . . . 5
82biimprd 226 . . . . . . . . 9
98expimpd 606 . . . . . . . 8
109ancomsd 455 . . . . . . 7
1110ad2antrr 730 . . . . . 6
1211rexlimdva 2924 . . . . 5
137, 12syl5 33 . . . 4
146, 13mpan2d 678 . . 3
1514ralrimdva 2850 . 2
165, 15impbid 193 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 187   wa 370   wceq 1437   wcel 1870  wral 2782  wrex 2783 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407 This theorem depends on definitions:  df-bi 188  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ral 2787  df-rex 2788  df-v 3089 This theorem is referenced by:  rexxfrd  4637  ralxfr2d  4638  ralxfr  4640  islindf4  19327  cmpfi  20354  rlimcnp  23756  ispisys2  28814  glbconN  32651  mapdordlem2  34914
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