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Theorem rexxfrd 4161
Description: Transfer universal quantification from a variable to another variable contained in expression . (Contributed by FL, 10-Apr-2007.) (Revised by Mario Carneiro, 15-Aug-2014.)
Hypotheses
Ref Expression
ralxfrd.1  C
ralxfrd.2  C
ralxfrd.3
Assertion
Ref Expression
rexxfrd  C
Distinct variable groups:   ,   ,,   , C   ,   ,,   ,
Allowed substitution hints:   ()   ()   ()    C()

Proof of Theorem rexxfrd
StepHypRef Expression
1 nfv 1418 . . . . 5  F/
2119.3 1443 . . . 4
3 ralxfrd.2 . . . . 5  C
4 df-rex 2306 . . . . . . . 8  C  C
5 19.29 1508 . . . . . . . . . 10  C  C
6 an12 495 . . . . . . . . . . 11  C  C
76exbii 1493 . . . . . . . . . 10  C  C
85, 7sylib 127 . . . . . . . . 9  C  C
9 df-rex 2306 . . . . . . . . 9  C  C
108, 9sylibr 137 . . . . . . . 8  C  C
114, 10sylan2b 271 . . . . . . 7  C  C
12 ralxfrd.3 . . . . . . . . . . 11
1312biimpd 132 . . . . . . . . . 10
1413expimpd 345 . . . . . . . . 9
1514ancomsd 256 . . . . . . . 8
1615reximdv 2414 . . . . . . 7  C  C
1711, 16syl5 28 . . . . . 6  C  C
1817adantr 261 . . . . 5  C  C
193, 18mpan2d 404 . . . 4  C
202, 19syl5bir 142 . . 3  C
2120rexlimdva 2427 . 2  C
22 ralxfrd.1 . . . 4  C
2312adantlr 446 . . . 4  C
2422, 23rspcedv 2654 . . 3  C
2524rexlimdva 2427 . 2  C
2621, 25impbid 120 1  C
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wb 98  wal 1240   wceq 1242  wex 1378   wcel 1390  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-bnd 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:  rexxfr2d  4163  rexxfr  4166
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