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Theorem sbcrextOLD 3409
 Description: Interchange class substitution and restricted existential quantifier. (Contributed by NM, 1-Mar-2008.) (Proof shortened by Mario Carneiro, 13-Oct-2016.) Obsolete as of 18-Aug-2018. (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
sbcrextOLD
Distinct variable groups:   ,   ,

Proof of Theorem sbcrextOLD
StepHypRef Expression
1 elex 3118 . 2
2 sbcng 3368 . . . . 5
32adantr 465 . . . 4
4 sbcralt 3408 . . . . . 6
5 nfnfc1 2622 . . . . . . . . 9
6 id 22 . . . . . . . . . 10
7 nfcvd 2620 . . . . . . . . . 10
86, 7nfeld 2627 . . . . . . . . 9
95, 8nfan1 1927 . . . . . . . 8
10 sbcng 3368 . . . . . . . . 9
1110adantl 466 . . . . . . . 8
129, 11ralbid 2891 . . . . . . 7
1312ancoms 453 . . . . . 6
144, 13bitrd 253 . . . . 5
1514notbid 294 . . . 4
163, 15bitrd 253 . . 3
17 dfrex2 2908 . . . 4
1817sbcbii 3387 . . 3
19 dfrex2 2908 . . 3
2016, 18, 193bitr4g 288 . 2
211, 20sylan 471 1
 Colors of variables: wff setvar class Syntax hints:  -.wn 3  ->wi 4  <->wb 184  /\wa 369  e.wcel 1818  F/_wnfc 2605  A.wral 2807  E.wrex 2808   cvv 3109  [.wsbc 3327 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-or 370  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-ral 2812  df-rex 2813  df-v 3111  df-sbc 3328
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