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Theorem sbcralt 3118
 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 3068 . 2
2 simpl 443 . . 3
3 sbsbc 3050 . . . . 5
4 nfcv 2489 . . . . . . 7
5 nfs1v 2106 . . . . . . 7
64, 5nfral 2667 . . . . . 6
7 sbequ12 1919 . . . . . . 7
87ralbidv 2634 . . . . . 6
96, 8sbie 2038 . . . . 5
103, 9bitr3i 242 . . . 4
11 nfnfc1 2492 . . . . . . 7
12 nfcvd 2490 . . . . . . . 8
13 id 19 . . . . . . . 8
1412, 13nfeqd 2503 . . . . . . 7
1511, 14nfan1 1881 . . . . . 6
16 dfsbcq2 3049 . . . . . . 7
1716adantl 452 . . . . . 6
1815, 17ralbid 2632 . . . . 5
1918adantll 694 . . . 4
2010, 19syl5bb 248 . . 3
212, 20sbcied 3082 . 2
221, 21syl5bbr 250 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 176   wa 358   wceq 1642  wsb 1648   wcel 1710  wnfc 2476  wral 2614  wsbc 3046 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619  df-v 2861  df-sbc 3047 This theorem is referenced by:  sbcrext  3119
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