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

Theorem intab 3723
 Description: The intersection of a special case of a class abstraction. may be free in and , which can be thought of a and . (Contributed by NM, 28-Jul-2006.) (Proof shortened by Mario Carneiro, 14-Nov-2016.)
Hypotheses
Ref Expression
intab.1
intab.2
Assertion
Ref Expression
intab
Distinct variable groups:   ,   ,   ,
Allowed substitution hints:   ()   ()

Proof of Theorem intab
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 eqeq1 2095 . . . . . . . . . 10
21anbi2d 453 . . . . . . . . 9
32exbidv 1754 . . . . . . . 8
43cbvabv 2212 . . . . . . 7
5 intab.2 . . . . . . 7
64, 5eqeltri 2161 . . . . . 6
7 nfe1 1431 . . . . . . . . 9
87nfab 2234 . . . . . . . 8
98nfeq2 2241 . . . . . . 7
10 eleq2 2152 . . . . . . . 8
1110imbi2d 229 . . . . . . 7
129, 11albid 1552 . . . . . 6
136, 12elab 2761 . . . . 5
14 19.8a 1528 . . . . . . . . 9
1514ex 114 . . . . . . . 8
1615alrimiv 1803 . . . . . . 7
17 intab.1 . . . . . . . 8
1817sbc6 2866 . . . . . . 7
1916, 18sylibr 133 . . . . . 6
20 df-sbc 2842 . . . . . 6
2119, 20sylib 121 . . . . 5
2213, 21mpgbir 1388 . . . 4
23 intss1 3709 . . . 4
2422, 23ax-mp 7 . . 3
25 19.29r 1558 . . . . . . . 8
26 simplr 498 . . . . . . . . . 10
27 pm3.35 340 . . . . . . . . . . 11
2827adantlr 462 . . . . . . . . . 10
2926, 28eqeltrd 2165 . . . . . . . . 9
3029exlimiv 1535 . . . . . . . 8
3125, 30syl 14 . . . . . . 7
3231ex 114 . . . . . 6
3332alrimiv 1803 . . . . 5
34 vex 2623 . . . . . 6
3534elintab 3705 . . . . 5
3633, 35sylibr 133 . . . 4
3736abssi 3097 . . 3
3824, 37eqssi 3042 . 2
3938, 4eqtri 2109 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 103  wal 1288   wceq 1290  wex 1427   wcel 1439  cab 2075  cvv 2620  wsbc 2841   wss 3000  cint 3694 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071 This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-v 2622  df-sbc 2842  df-in 3006  df-ss 3013  df-int 3695 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator