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Theorem ralxfrd 4194
 Description: Transfer universal quantification from a variable to another variable contained in expression . (Contributed by NM, 15-Aug-2014.) (Proof shortened by Mario Carneiro, 19-Nov-2016.)
Hypotheses
Ref Expression
ralxfrd.1
ralxfrd.2
ralxfrd.3
Assertion
Ref Expression
ralxfrd
Distinct variable groups:   ,   ,,   ,   ,   ,,   ,
Allowed substitution hints:   ()   ()   ()   ()

Proof of Theorem ralxfrd
StepHypRef Expression
1 ralxfrd.1 . . . 4
2 ralxfrd.3 . . . . 5
32adantlr 446 . . . 4
41, 3rspcdv 2659 . . 3
54ralrimdva 2399 . 2
6 ralxfrd.2 . . . 4
7 r19.29 2450 . . . . 5
82biimprd 147 . . . . . . . . 9
98expimpd 345 . . . . . . . 8
109ancomsd 256 . . . . . . 7
1110ad2antrr 457 . . . . . 6
1211rexlimdva 2433 . . . . 5
137, 12syl5 28 . . . 4
146, 13mpan2d 404 . . 3
1514ralrimdva 2399 . 2
165, 15impbid 120 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 97   wb 98   wceq 1243   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:  ralxfr2d  4196  ralxfr  4198
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