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Theorem sbcrext 3119
 Description: Interchange class substitution and restricted existential quantifier. (Contributed by NM, 1-Mar-2008.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)
Assertion
Ref Expression
sbcrext
Distinct variable groups:   ,   ,
Allowed substitution hints:   (,)   (,)   ()   (,)

Proof of Theorem sbcrext
StepHypRef Expression
1 elex 2867 . 2
2 sbcng 3086 . . . . 5
32adantr 451 . . . 4
4 sbcralt 3118 . . . . . 6
5 nfnfc1 2492 . . . . . . . . 9
6 id 19 . . . . . . . . . 10
7 nfcvd 2490 . . . . . . . . . 10
86, 7nfeld 2504 . . . . . . . . 9
95, 8nfan1 1881 . . . . . . . 8
10 sbcng 3086 . . . . . . . . 9
1110adantl 452 . . . . . . . 8
129, 11ralbid 2632 . . . . . . 7
1312ancoms 439 . . . . . 6
144, 13bitrd 244 . . . . 5
1514notbid 285 . . . 4
163, 15bitrd 244 . . 3
17 dfrex2 2627 . . . 4
1817sbcbii 3101 . . 3
19 dfrex2 2627 . . 3
2016, 18, 193bitr4g 279 . 2
211, 20sylan 457 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wb 176   wa 358   wcel 1710  wnfc 2476  wral 2614  wrex 2615  cvv 2859  wsbc 3046 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619  df-rex 2620  df-v 2861  df-sbc 3047 This theorem is referenced by: (None)
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