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Theorem rexxfr2d 4163
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  C  V
ralxfr2d.2  C
ralxfr2d.3
Assertion
Ref Expression
rexxfr2d  C
Distinct variable groups:   ,   ,,   , C   ,   ,,   ,
Allowed substitution hints:   ()   ()   ()    C()    V(,)

Proof of Theorem rexxfr2d
StepHypRef Expression
1 ralxfr2d.1 . . . 4  C  V
2 elisset 2562 . . . 4  V
31, 2syl 14 . . 3  C
4 ralxfr2d.2 . . . . . . . 8  C
54biimprd 147 . . . . . . 7  C
6 r19.23v 2419 . . . . . . 7  C  C
75, 6sylibr 137 . . . . . 6  C
87r19.21bi 2401 . . . . 5  C
9 eleq1 2097 . . . . 5
108, 9mpbidi 140 . . . 4  C
1110exlimdv 1697 . . 3  C
123, 11mpd 13 . 2  C
134biimpa 280 . 2  C
14 ralxfr2d.3 . 2
1512, 13, 14rexxfrd 4161 1  C
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wb 98   wceq 1242  wex 1378   wcel 1390  wral 2300  wrex 2301
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553
This theorem is referenced by:  rexrn  5247  rexima  5337
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