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Theorem rexxfr2d 4354
 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 2672 . . . 4
31, 2syl 14 . . 3
4 ralxfr2d.2 . . . . . . . 8
54biimprd 157 . . . . . . 7
6 r19.23v 2516 . . . . . . 7
75, 6sylibr 133 . . . . . 6
87r19.21bi 2495 . . . . 5
9 eleq1 2178 . . . . 5
108, 9mpbidi 150 . . . 4
1110exlimdv 1773 . . 3
123, 11mpd 13 . 2
134biimpa 292 . 2
14 ralxfr2d.3 . 2
1512, 13, 14rexxfrd 4352 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 103   wb 104   wceq 1314  wex 1451   wcel 1463  wral 2391  wrex 2392 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097 This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660 This theorem is referenced by:  rexrn  5523  rexima  5622  cnptopresti  12302  cnptoprest  12303  txrest  12340
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