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Theorem ralxfr2d 4731
 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 2958 . . . 4
31, 2syl 16 . . 3
4 ralxfr2d.2 . . . . . . . 8
54biimprd 215 . . . . . . 7
6 r19.23v 2814 . . . . . . 7
75, 6sylibr 204 . . . . . 6
87r19.21bi 2796 . . . . 5
9 eleq1 2495 . . . . 5
108, 9mpbidi 208 . . . 4
1110exlimdv 1646 . . 3
123, 11mpd 15 . 2
134biimpa 471 . 2
14 ralxfr2d.3 . 2
1512, 13, 14ralxfrd 4729 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wa 359  wex 1550   wceq 1652   wcel 1725  wral 2697  wrex 2698 This theorem is referenced by:  rexxfr2d  4732  ralrn  5865  ralima  5970  cnrest2  17342  cnprest2  17346  consuba  17475  subislly  17536  trfbas2  17867  trfil2  17911  flimrest  18007  fclsrest  18048  tsmssubm  18164 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 2416 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 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-rex 2703  df-v 2950
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