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Theorem intab 4109
 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 6037 or abexssex 6038 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 2449 . . . . . . . . . 10
21anbi2d 686 . . . . . . . . 9
32exbidv 1638 . . . . . . . 8
43cbvabv 2562 . . . . . . 7
5 intab.2 . . . . . . 7
64, 5eqeltri 2513 . . . . . 6
7 nfe1 1750 . . . . . . . . 9
87nfab 2583 . . . . . . . 8
98nfeq2 2590 . . . . . . 7
10 eleq2 2504 . . . . . . . 8
1110imbi2d 309 . . . . . . 7
129, 11albid 1791 . . . . . 6
136, 12elab 3091 . . . . 5
14 19.8a 1765 . . . . . . . . 9
1514ex 425 . . . . . . . 8
1615alrimiv 1643 . . . . . . 7
17 intab.1 . . . . . . . 8
1817sbc6 3196 . . . . . . 7
1916, 18sylibr 205 . . . . . 6
20 df-sbc 3171 . . . . . 6
2119, 20sylib 190 . . . . 5
2213, 21mpgbir 1560 . . . 4
23 intss1 4094 . . . 4
2422, 23ax-mp 5 . . 3
25 19.29r 1609 . . . . . . . 8
26 simplr 733 . . . . . . . . . 10
27 pm3.35 572 . . . . . . . . . . 11
2827adantlr 697 . . . . . . . . . 10
2926, 28eqeltrd 2517 . . . . . . . . 9
3029exlimiv 1646 . . . . . . . 8
3125, 30syl 16 . . . . . . 7
3231ex 425 . . . . . 6
3332alrimiv 1643 . . . . 5
34 vex 2968 . . . . . 6
3534elintab 4090 . . . . 5
3633, 35sylibr 205 . . . 4
3736abssi 3407 . . 3
3824, 37eqssi 3353 . 2
3938, 4eqtri 2463 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 360  wal 1550  wex 1551   wceq 1654   wcel 1728  cab 2429  cvv 2965  wsbc 3170   wss 3309  cint 4079 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1628  ax-9 1669  ax-8 1690  ax-6 1747  ax-7 1752  ax-11 1764  ax-12 1954  ax-ext 2424 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1661  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-v 2967  df-sbc 3171  df-in 3316  df-ss 3323  df-int 4080
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