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Theorem rexxfr2d 4197
 Description: Transfer universal quantification from a variable to another variable contained in expression . (Contributed by Mario Carneiro, 20-Aug-2014.) (Proof shortened by Mario Carneiro, 19-Nov-2016.)
Hypotheses
Ref Expression
ralxfr2d.1
ralxfr2d.2
ralxfr2d.3
Assertion
Ref Expression
rexxfr2d
Distinct variable groups:   ,   ,,   ,   ,   ,,   ,
Allowed substitution hints:   ()   ()   ()   ()   (,)

Proof of Theorem rexxfr2d
StepHypRef Expression
1 ralxfr2d.1 . . . 4
2 elisset 2568 . . . 4
31, 2syl 14 . . 3
4 ralxfr2d.2 . . . . . . . 8
54biimprd 147 . . . . . . 7
6 r19.23v 2425 . . . . . . 7
75, 6sylibr 137 . . . . . 6
87r19.21bi 2407 . . . . 5
9 eleq1 2100 . . . . 5
108, 9mpbidi 140 . . . 4
1110exlimdv 1700 . . 3
123, 11mpd 13 . 2
134biimpa 280 . 2
14 ralxfr2d.3 . 2
1512, 13, 14rexxfrd 4195 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 97   wb 98   wceq 1243  wex 1381   wcel 1393  wral 2306  wrex 2307 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559 This theorem is referenced by:  rexrn  5304  rexima  5394
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