Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  sbcralt Unicode version

Theorem sbcralt 2990
 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 2935 . 2
2 simpl 108 . . 3
3 sbsbc 2918 . . . . 5
4 nfcv 2282 . . . . . . 7
5 nfs1v 1913 . . . . . . 7
64, 5nfralxy 2475 . . . . . 6
7 sbequ12 1745 . . . . . . 7
87ralbidv 2439 . . . . . 6
96, 8sbie 1765 . . . . 5
103, 9bitr3i 185 . . . 4
11 nfnfc1 2285 . . . . . . 7
12 nfcvd 2283 . . . . . . . 8
13 id 19 . . . . . . . 8
1412, 13nfeqd 2297 . . . . . . 7
1511, 14nfan1 1544 . . . . . 6
16 dfsbcq2 2917 . . . . . . 7
1716adantl 275 . . . . . 6
1815, 17ralbid 2437 . . . . 5
1918adantll 468 . . . 4
2010, 19syl5bb 191 . . 3
212, 20sbcied 2950 . 2
221, 21bitr3id 193 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 103   wb 104   wceq 1332   wcel 1481  wsb 1736  wnfc 2269  wral 2417  wsbc 2914 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-v 2692  df-sbc 2915 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator