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Theorem sbcrext 3407
 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:   ,   ,
Allowed substitution hints:   (,)   (,)   ()

Proof of Theorem sbcrext
StepHypRef Expression
1 sbcng 3368 . . . . 5
21adantr 465 . . . 4
3 sbcralt 3406 . . . . . 6
4 nfnfc1 2622 . . . . . . . . 9
5 id 22 . . . . . . . . . 10
6 nfcvd 2620 . . . . . . . . . 10
75, 6nfeld 2627 . . . . . . . . 9
84, 7nfan1 1928 . . . . . . . 8
9 sbcng 3368 . . . . . . . . 9
109adantl 466 . . . . . . . 8
118, 10ralbid 2891 . . . . . . 7
1211ancoms 453 . . . . . 6
133, 12bitrd 253 . . . . 5
1413notbid 294 . . . 4
152, 14bitrd 253 . . 3
16 dfrex2 2908 . . . 4
1716sbcbii 3387 . . 3
18 dfrex2 2908 . . 3
1915, 17, 183bitr4g 288 . 2
20 sbcex 3337 . . . . 5
2120con3i 135 . . . 4
2221adantr 465 . . 3
23 sbcex 3337 . . . . . . 7
2423a1ii 27 . . . . . 6
254, 7, 24rexlimd2 2940 . . . . 5
2625con3rr3 136 . . . 4
2726imp 429 . . 3
2822, 272falsed 351 . 2
2919, 28pm2.61ian 790 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wb 184   wa 369   wcel 1819  wnfc 2605  wral 2807  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 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  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 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812  df-rex 2813  df-v 3111  df-sbc 3328 This theorem is referenced by:  sbcrex  3409
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