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Theorem ralxfrd2 39008
 Description: Transfer universal quantification from a variable to another variable contained in expression . Variant of ralxfrd 4614. (Contributed by Alexander van der Vekens, 25-Apr-2018.)
Hypotheses
Ref Expression
ralxfrd2.1
ralxfrd2.2
ralxfrd2.3
Assertion
Ref Expression
ralxfrd2
Distinct variable groups:   ,   ,,   ,   ,   ,,   ,
Allowed substitution hints:   ()   ()   ()   ()

Proof of Theorem ralxfrd2
StepHypRef Expression
1 ralxfrd2.1 . . . 4
2 ralxfrd2.3 . . . . 5
323expa 1208 . . . 4
41, 3rspcdv 3153 . . 3
54ralrimdva 2806 . 2
6 ralxfrd2.2 . . . 4
7 r19.29 2925 . . . . 5
8 simplll 768 . . . . . . . . . 10
9 simplr 762 . . . . . . . . . 10
10 simpr 463 . . . . . . . . . 10
118, 9, 10, 2syl3anc 1268 . . . . . . . . 9
1211exbiri 628 . . . . . . . 8
1312com23 81 . . . . . . 7
1413impd 433 . . . . . 6
1514rexlimdva 2879 . . . . 5
167, 15syl5 33 . . . 4
176, 16mpan2d 680 . . 3
1817ralrimdva 2806 . 2
195, 18impbid 194 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 188   wa 371   w3a 985   wceq 1444   wcel 1887  wral 2737  wrex 2738 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431 This theorem depends on definitions:  df-bi 189  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ral 2742  df-rex 2743  df-v 3047 This theorem is referenced by:  rexxfrd2  39009
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