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Theorem elinintrab 36254
 Description: Two ways of saying a set is an element of the intersection of a class with the intersection of a class. (Contributed by RP, 14-Aug-2020.)
Assertion
Ref Expression
elinintrab
Distinct variable groups:   ,   ,,   ,,
Allowed substitution hints:   ()   (,)

Proof of Theorem elinintrab
StepHypRef Expression
1 vex 3034 . . . 4
21inex2 4538 . . 3
3 inss1 3643 . . 3
42, 3elmapintrab 36253 . 2
5 elin 3608 . . . . . . . 8
65imbi2i 319 . . . . . . 7
7 jcab 880 . . . . . . 7
86, 7bitri 257 . . . . . 6
98albii 1699 . . . . 5
10 19.26 1740 . . . . . 6
11 19.23v 1826 . . . . . . 7
1211anbi1i 709 . . . . . 6
1310, 12bitri 257 . . . . 5
149, 13bitri 257 . . . 4
1514anbi2i 708 . . 3
16 anabs5 826 . . 3
1715, 16bitri 257 . 2
184, 17syl6bb 269 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 189   wa 376  wal 1450   wceq 1452  wex 1671   wcel 1904  crab 2760   cin 3389  cpw 3942  cint 4226 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518 This theorem depends on definitions:  df-bi 190  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ral 2761  df-rab 2765  df-v 3033  df-in 3397  df-ss 3404  df-pw 3944  df-int 4227 This theorem is referenced by:  inintabss  36255  inintabd  36256
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