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Theorem sbcrext 3373
 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 3340 . . . . 5
21adantr 466 . . . 4
3 sbcralt 3372 . . . . . 6
4 nfnfc1 2583 . . . . . . . . 9
5 id 22 . . . . . . . . . 10
6 nfcvd 2581 . . . . . . . . . 10
75, 6nfeld 2588 . . . . . . . . 9
84, 7nfan1 1987 . . . . . . . 8
9 sbcng 3340 . . . . . . . . 9
109adantl 467 . . . . . . . 8
118, 10ralbid 2856 . . . . . . 7
1211ancoms 454 . . . . . 6
133, 12bitrd 256 . . . . 5
1413notbid 295 . . . 4
152, 14bitrd 256 . . 3
16 dfrex2 2873 . . . 4
1716sbcbii 3355 . . 3
18 dfrex2 2873 . . 3
1915, 17, 183bitr4g 291 . 2
20 sbcex 3309 . . . . 5
2120con3i 140 . . . 4
23 sbcex 3309 . . . . . . 7
24232a1i 12 . . . . . 6
254, 7, 24rexlimd2 2905 . . . . 5
2625con3rr3 141 . . . 4
2726imp 430 . . 3
2822, 272falsed 352 . 2
2919, 28pm2.61ian 797 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wb 187   wa 370   wcel 1872  wnfc 2566  wral 2771  wrex 2772  cvv 3080  wsbc 3299 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401 This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ral 2776  df-rex 2777  df-v 3082  df-sbc 3300 This theorem is referenced by:  sbcrex  3375
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