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Theorem sbcrextOLD 3408
 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:   ,   ,
Allowed substitution hints:   (,)   (,)   ()   (,)

Proof of Theorem sbcrextOLD
StepHypRef Expression
1 elex 3117 . 2
2 sbcng 3367 . . . . 5
32adantr 465 . . . 4
4 sbcralt 3407 . . . . . 6
5 nfnfc1 2627 . . . . . . . . 9
6 id 22 . . . . . . . . . 10
7 nfcvd 2625 . . . . . . . . . 10
86, 7nfeld 2632 . . . . . . . . 9
95, 8nfan1 1869 . . . . . . . 8
10 sbcng 3367 . . . . . . . . 9
1110adantl 466 . . . . . . . 8
129, 11ralbid 2893 . . . . . . 7
1312ancoms 453 . . . . . 6
144, 13bitrd 253 . . . . 5
1514notbid 294 . . . 4
163, 15bitrd 253 . . 3
17 dfrex2 2910 . . . 4
1817sbcbii 3386 . . 3
19 dfrex2 2910 . . 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   wcel 1762  wnfc 2610  wral 2809  wrex 2810  cvv 3108  wsbc 3326 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ral 2814  df-rex 2815  df-v 3110  df-sbc 3327 This theorem is referenced by: (None)
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