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Theorem intab 4317
 Description: The intersection of a special case of a class abstraction. may be free in and , which can be thought of a ( ) and A( ). Typically, abrexex2 6781 or abexssex 6782 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:   ,   ,   ,

Proof of Theorem intab
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 eqeq1 2461 . . . . . . . . . 10
21anbi2d 703 . . . . . . . . 9
32exbidv 1714 . . . . . . . 8
43cbvabv 2600 . . . . . . 7
5 intab.2 . . . . . . 7
64, 5eqeltri 2541 . . . . . 6
7 nfe1 1840 . . . . . . . . 9
87nfab 2623 . . . . . . . 8
98nfeq2 2636 . . . . . . 7
10 eleq2 2530 . . . . . . . 8
1110imbi2d 316 . . . . . . 7
129, 11albid 1885 . . . . . 6
136, 12elab 3246 . . . . 5
14 19.8a 1857 . . . . . . . . 9
1514ex 434 . . . . . . . 8
1615alrimiv 1719 . . . . . . 7
17 intab.1 . . . . . . . 8
1817sbc6 3354 . . . . . . 7
1916, 18sylibr 212 . . . . . 6
20 df-sbc 3328 . . . . . 6
2119, 20sylib 196 . . . . 5
2213, 21mpgbir 1622 . . . 4
23 intss1 4301 . . . 4
2422, 23ax-mp 5 . . 3
25 19.29r 1684 . . . . . . . 8
26 simplr 755 . . . . . . . . . 10
27 pm3.35 587 . . . . . . . . . . 11
2827adantlr 714 . . . . . . . . . 10
2926, 28eqeltrd 2545 . . . . . . . . 9
3029exlimiv 1722 . . . . . . . 8
3125, 30syl 16 . . . . . . 7
3231ex 434 . . . . . 6
3332alrimiv 1719 . . . . 5
34 vex 3112 . . . . . 6
3534elintab 4297 . . . . 5
3633, 35sylibr 212 . . . 4
3736abssi 3574 . . 3
3824, 37eqssi 3519 . 2
3938, 4eqtri 2486 1
 Colors of variables: wff setvar class Syntax hints:  ->wi 4  /\wa 369  A.wal 1393  =wceq 1395  E.wex 1612  e.wcel 1818  {cab 2442   cvv 3109  [.wsbc 3327  C_wss 3475  |^|cint 4286 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-v 3111  df-sbc 3328  df-in 3482  df-ss 3489  df-int 4287
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