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Theorem ralxfr2d 4606
 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:   ,   ,,   ,   ,   ,,   ,
Allowed substitution hints:   ()   ()   ()   ()   (,)

Proof of Theorem ralxfr2d
StepHypRef Expression
1 ralxfr2d.1 . . . 4
2 elisset 3069 . . . 4
31, 2syl 17 . . 3
4 ralxfr2d.2 . . . . . . . 8
54biimprd 223 . . . . . . 7
6 r19.23v 2883 . . . . . . 7
75, 6sylibr 212 . . . . . 6
87r19.21bi 2772 . . . . 5
9 eleq1 2474 . . . . 5
108, 9mpbidi 216 . . . 4
1110exlimdv 1745 . . 3
123, 11mpd 15 . 2
134biimpa 482 . 2
14 ralxfr2d.3 . 2
1512, 13, 14ralxfrd 4604 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 184   wa 367   wceq 1405  wex 1633   wcel 1842  wral 2753  wrex 2754 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380 This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ral 2758  df-rex 2759  df-v 3060 This theorem is referenced by:  rexxfr2d  4607  ralrn  6011  ralima  6132  cnrest2  20078  cnprest2  20082  consuba  20211  subislly  20272  trfbas2  20634  trfil2  20678  flimrest  20774  fclsrest  20815  tsmssubm  20934  metucn  21382  extoimad  35972
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