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Theorem sbcrext 2808
Description: Interchange class substitution and restricted existential quantifier. (Contributed by NM, 1-Mar-2008.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)
Assertion
Ref Expression
sbcrext  V  F/_  [.  ].  [.  ].
Distinct variable groups:   ,   ,
Allowed substitution hints:   (,)   (,)   ()    V(,)

Proof of Theorem sbcrext
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 sbcco 2758 . 2  [.  ]. [.  ].  [.  ].
2 simpl 102 . . 3  V  F/_  V
3 sbsbc 2741 . . . . 5  [.  ].
4 nfcv 2156 . . . . . . 7  F/_
5 nfs1v 1793 . . . . . . 7  F/
64, 5nfrexxy 2335 . . . . . 6  F/
7 sbequ12 1632 . . . . . . 7
87rexbidv 2301 . . . . . 6
96, 8sbie 1652 . . . . 5
103, 9bitr3i 175 . . . 4  [.  ].
11 nfnfc1 2159 . . . . . . 7  F/
F/_
12 nfcvd 2157 . . . . . . . 8  F/_  F/_
13 id 19 . . . . . . . 8  F/_  F/_
1412, 13nfeqd 2170 . . . . . . 7  F/_  F/
1511, 14nfan1 1434 . . . . . 6  F/ F/_
16 dfsbcq2 2740 . . . . . . 7  [.  ].
1716adantl 262 . . . . . 6 
F/_  [.  ].
1815, 17rexbid 2299 . . . . 5 
F/_  [.  ].
1918adantll 448 . . . 4  V  F/_  [.  ].
2010, 19syl5bb 181 . . 3  V  F/_  [.  ].  [.  ].
212, 20sbcied 2772 . 2  V  F/_  [.  ]. [.  ].  [.  ].
221, 21syl5bbr 183 1  V  F/_  [.  ].  [.  ].
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wb 98   wceq 1226   wcel 1370  wsb 1623   F/_wnfc 2143  wrex 2281   [.wsbc 2737
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 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000
This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-rex 2286  df-v 2533  df-sbc 2738
This theorem is referenced by: (None)
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