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Theorem sbcrext 3380
Description: Interchange class substitution and restricted existential quantifier. (Contributed by NM, 1-Mar-2008.) (Proof shortened by Mario Carneiro, 13-Oct-2016.) (Revised by NM, 18-Aug-2018.)
Assertion
Ref Expression
sbcrext
Distinct variable groups:   ,   ,

Proof of Theorem sbcrext
StepHypRef Expression
1 sbcng 3338 . . . . 5
21adantr 465 . . . 4
3 sbcralt 3378 . . . . . 6
4 nfnfc1 2619 . . . . . . . . 9
5 id 22 . . . . . . . . . 10
6 nfcvd 2617 . . . . . . . . . 10
75, 6nfeld 2624 . . . . . . . . 9
84, 7nfan1 1865 . . . . . . . 8
9 sbcng 3338 . . . . . . . . 9
109adantl 466 . . . . . . . 8
118, 10ralbid 2844 . . . . . . 7
1211ancoms 453 . . . . . 6
133, 12bitrd 253 . . . . 5
1413notbid 294 . . . 4
152, 14bitrd 253 . . 3
16 dfrex2 2861 . . . 4
1716sbcbii 3357 . . 3
18 dfrex2 2861 . . 3
1915, 17, 183bitr4g 288 . 2
20 sbcex 3307 . . . . 5
2120con3i 135 . . . 4
2221adantr 465 . . 3
23 sbcex 3307 . . . . . . 7
2423a1ii 27 . . . . . 6
254, 7, 24rexlimd2 2948 . . . . 5
2625con3rr3 136 . . . 4
2726imp 429 . . 3
2822, 272falsed 351 . 2
2919, 28pm2.61ian 788 1
Colors of variables: wff setvar class
Syntax hints:  -.wn 3  ->wi 4  <->wb 184  /\wa 369  e.wcel 1758  F/_wnfc 2602  A.wral 2800  E.wrex 2801   cvv 3081  [.wsbc 3297
This theorem is referenced by:  sbcrex  3382
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ral 2805  df-rex 2806  df-v 3083  df-sbc 3298
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