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Theorem rspc3ev 3223
Description: 3-variable restricted existential specialization, using implicit substitution. (Contributed by NM, 25-Jul-2012.)
Hypotheses
Ref Expression
rspc3v.1
rspc3v.2
rspc3v.3
Assertion
Ref Expression
rspc3ev
Distinct variable groups:   ,   ,   ,   , , ,   , ,   ,   ,   ,S,   , , ,

Proof of Theorem rspc3ev
StepHypRef Expression
1 simpl1 999 . 2
2 simpl2 1000 . 2
3 rspc3v.3 . . . 4
43rspcev 3210 . . 3
543ad2antl3 1160 . 2
6 rspc3v.1 . . . 4
76rexbidv 2968 . . 3
8 rspc3v.2 . . . 4
98rexbidv 2968 . . 3
107, 9rspc2ev 3221 . 2
111, 2, 5, 10syl3anc 1228 1
Colors of variables: wff setvar class
Syntax hints:  ->wi 4  <->wb 184  /\wa 369  /\w3a 973  =wceq 1395  e.wcel 1818  E.wrex 2808
This theorem is referenced by:  f1dom3el3dif  6176  pmltpclem1  21860  axlowdim  24264  axeuclidlem  24265  br8d  27463  br8  29185  br6  29186  jm2.27  30950  3dim1lem5  35190  lplni2  35261
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-rex 2813  df-v 3111
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