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Theorem sbcralt 3235
 Description: Interchange class substitution and restricted quantifier. (Contributed by NM, 1-Mar-2008.) (Revised by David Abernethy, 22-Feb-2010.)
Assertion
Ref Expression
sbcralt
Distinct variable groups:   ,   ,
Allowed substitution hints:   (,)   (,)   ()   (,)

Proof of Theorem sbcralt
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 sbcco 3185 . 2
2 simpl 445 . . 3
3 sbsbc 3167 . . . . 5
4 nfcv 2574 . . . . . . 7
5 nfs1v 2184 . . . . . . 7
64, 5nfral 2761 . . . . . 6
7 sbequ12 1945 . . . . . . 7
87ralbidv 2727 . . . . . 6
96, 8sbie 2152 . . . . 5
103, 9bitr3i 244 . . . 4
11 nfnfc1 2577 . . . . . . 7
12 nfcvd 2575 . . . . . . . 8
13 id 21 . . . . . . . 8
1412, 13nfeqd 2588 . . . . . . 7
1511, 14nfan1 1846 . . . . . 6
16 dfsbcq2 3166 . . . . . . 7
1716adantl 454 . . . . . 6
1815, 17ralbid 2725 . . . . 5
1918adantll 696 . . . 4
2010, 19syl5bb 250 . . 3
212, 20sbcied 3199 . 2
221, 21syl5bbr 252 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 178   wa 360   wceq 1653  wsb 1659   wcel 1726  wnfc 2561  wral 2707  wsbc 3163 This theorem is referenced by:  sbcrext  3236 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-v 2960  df-sbc 3164
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