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Theorem ralxfrd 4160
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  C
ralxfrd.2  C
ralxfrd.3
Assertion
Ref Expression
ralxfrd  C
Distinct variable groups:   ,   ,,   , C   ,   ,,   ,
Allowed substitution hints:   ()   ()   ()    C()

Proof of Theorem ralxfrd
StepHypRef Expression
1 ralxfrd.1 . . . 4  C
2 ralxfrd.3 . . . . 5
32adantlr 446 . . . 4  C
41, 3rspcdv 2653 . . 3  C
54ralrimdva 2393 . 2  C
6 ralxfrd.2 . . . 4  C
7 r19.29 2444 . . . . 5  C  C  C
82biimprd 147 . . . . . . . . 9
98expimpd 345 . . . . . . . 8
109ancomsd 256 . . . . . . 7
1110ad2antrr 457 . . . . . 6  C
1211rexlimdva 2427 . . . . 5  C
137, 12syl5 28 . . . 4  C  C
146, 13mpan2d 404 . . 3  C
1514ralrimdva 2393 . 2  C
165, 15impbid 120 1  C
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wb 98   wceq 1242   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-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:  ralxfr2d  4162  ralxfr  4164
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