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Theorem sbcralt 2828
Description: Interchange class substitution and restricted quantifier. (Contributed by NM, 1-Mar-2008.) (Revised by David Abernethy, 22-Feb-2010.)
Assertion
Ref Expression
sbcralt  V  F/_  [.  ].  [.  ].
Distinct variable groups:   ,   ,
Allowed substitution hints:   (,)   (,)   ()    V(,)

Proof of Theorem sbcralt
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 sbcco 2779 . 2  [.  ]. [.  ].  [.  ].
2 simpl 102 . . 3  V  F/_  V
3 sbsbc 2762 . . . . 5  [.  ].
4 nfcv 2175 . . . . . . 7  F/_
5 nfs1v 1812 . . . . . . 7  F/
64, 5nfralxy 2354 . . . . . 6  F/
7 sbequ12 1651 . . . . . . 7
87ralbidv 2320 . . . . . 6
96, 8sbie 1671 . . . . 5
103, 9bitr3i 175 . . . 4  [.  ].
11 nfnfc1 2178 . . . . . . 7  F/
F/_
12 nfcvd 2176 . . . . . . . 8  F/_  F/_
13 id 19 . . . . . . . 8  F/_  F/_
1412, 13nfeqd 2189 . . . . . . 7  F/_  F/
1511, 14nfan1 1453 . . . . . 6  F/ F/_
16 dfsbcq2 2761 . . . . . . 7  [.  ].
1716adantl 262 . . . . . 6 
F/_  [.  ].
1815, 17ralbid 2318 . . . . 5 
F/_  [.  ].
1918adantll 445 . . . 4  V  F/_  [.  ].
2010, 19syl5bb 181 . . 3  V  F/_  [.  ].  [.  ].
212, 20sbcied 2793 . 2  V  F/_  [.  ]. [.  ].  [.  ].
221, 21syl5bbr 183 1  V  F/_  [.  ].  [.  ].
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wb 98   wceq 1242   wcel 1390  wsb 1642   F/_wnfc 2162  wral 2300   [.wsbc 2758
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-v 2553  df-sbc 2759
This theorem is referenced by: (None)
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