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Theorem sbcralg 2864
 Description: Interchange class substitution and restricted quantifier. (Contributed by NM, 15-Nov-2005.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
Assertion
Ref Expression
sbcralg
Distinct variable groups:   ,   ,   ,
Allowed substitution hints:   (,)   ()   ()   (,)

Proof of Theorem sbcralg
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 dfsbcq2 2790 . 2
2 dfsbcq2 2790 . . 3
32ralbidv 2343 . 2
4 nfcv 2194 . . . 4
5 nfs1v 1831 . . . 4
64, 5nfralxy 2377 . . 3
7 sbequ12 1670 . . . 4
87ralbidv 2343 . . 3
96, 8sbie 1690 . 2
101, 3, 9vtoclbg 2631 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 102   wceq 1259   wcel 1409  wsb 1661  wral 2323  wsbc 2787 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038 This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-v 2576  df-sbc 2788 This theorem is referenced by:  r19.12sn  3464
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