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Theorem intab 3956
 Description: The intersection of a special case of a class abstraction. may be free in and , which can be thought of a and . Typically, abrexex2 in set.mm or abexssex in set.mm can be used to satisfy the second hypothesis. (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 2359 . . . . . . . . . 10
21anbi2d 684 . . . . . . . . 9
32exbidv 1626 . . . . . . . 8
43cbvabv 2472 . . . . . . 7
5 intab.2 . . . . . . 7
64, 5eqeltri 2423 . . . . . 6
7 nfe1 1732 . . . . . . . . 9
87nfab 2493 . . . . . . . 8
98nfeq2 2500 . . . . . . 7
10 eleq2 2414 . . . . . . . 8
1110imbi2d 307 . . . . . . 7
129, 11albid 1772 . . . . . 6
136, 12elab 2985 . . . . 5
14 19.8a 1756 . . . . . . . . 9
1514ex 423 . . . . . . . 8
1615alrimiv 1631 . . . . . . 7
17 intab.1 . . . . . . . 8
1817sbc6 3072 . . . . . . 7
1916, 18sylibr 203 . . . . . 6
20 df-sbc 3047 . . . . . 6
2119, 20sylib 188 . . . . 5
2213, 21mpgbir 1550 . . . 4
23 intss1 3941 . . . 4
2422, 23ax-mp 8 . . 3
25 19.29r 1597 . . . . . . . 8
26 simplr 731 . . . . . . . . . 10
27 pm3.35 570 . . . . . . . . . . 11
2827adantlr 695 . . . . . . . . . 10
2926, 28eqeltrd 2427 . . . . . . . . 9
3029exlimiv 1634 . . . . . . . 8
3125, 30syl 15 . . . . . . 7
3231ex 423 . . . . . 6
3332alrimiv 1631 . . . . 5
34 vex 2862 . . . . . 6
3534elintab 3937 . . . . 5
3633, 35sylibr 203 . . . 4
3736abssi 3341 . . 3
3824, 37eqssi 3288 . 2
3938, 4eqtri 2373 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 358  wal 1540  wex 1541   wceq 1642   wcel 1710  cab 2339  cvv 2859  wsbc 3046   wss 3257  cint 3926 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259  df-int 3927 This theorem is referenced by: (None)
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