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Theorem dfiin2g 4302
 Description: Alternate definition of indexed intersection when is a set. (Contributed by Jeff Hankins, 27-Aug-2009.)
Assertion
Ref Expression
dfiin2g
Distinct variable groups:   ,   ,   ,
Allowed substitution hints:   ()   ()   (,)

Proof of Theorem dfiin2g
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ral 2761 . . . 4
2 df-ral 2761 . . . . . 6
3 eleq2 2538 . . . . . . . . . . . . 13
43biimprcd 233 . . . . . . . . . . . 12
54alrimiv 1781 . . . . . . . . . . 11
6 eqid 2471 . . . . . . . . . . . 12
7 eqeq1 2475 . . . . . . . . . . . . . 14
87, 3imbi12d 327 . . . . . . . . . . . . 13
98spcgv 3120 . . . . . . . . . . . 12
106, 9mpii 43 . . . . . . . . . . 11
115, 10impbid2 209 . . . . . . . . . 10
1211imim2i 16 . . . . . . . . 9
1312pm5.74d 255 . . . . . . . 8
1413alimi 1692 . . . . . . 7
15 albi 1698 . . . . . . 7
1614, 15syl 17 . . . . . 6
172, 16sylbi 200 . . . . 5
18 df-ral 2761 . . . . . . . 8
1918albii 1699 . . . . . . 7
20 alcom 1940 . . . . . . 7
2119, 20bitr4i 260 . . . . . 6
22 r19.23v 2863 . . . . . . . 8
23 vex 3034 . . . . . . . . . 10
24 eqeq1 2475 . . . . . . . . . . 11
2524rexbidv 2892 . . . . . . . . . 10
2623, 25elab 3173 . . . . . . . . 9
2726imbi1i 332 . . . . . . . 8
2822, 27bitr4i 260 . . . . . . 7
2928albii 1699 . . . . . 6
30 19.21v 1794 . . . . . . 7
3130albii 1699 . . . . . 6
3221, 29, 313bitr3ri 284 . . . . 5
3317, 32syl6bb 269 . . . 4
341, 33syl5bb 265 . . 3
3534abbidv 2589 . 2
36 df-iin 4272 . 2
37 df-int 4227 . 2
3835, 36, 373eqtr4g 2530 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 189  wal 1450   wceq 1452   wcel 1904  cab 2457  wral 2756  wrex 2757  cint 4226  ciin 4270 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 This theorem depends on definitions:  df-bi 190  df-an 378  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-rex 2762  df-v 3033  df-int 4227  df-iin 4272 This theorem is referenced by:  dfiin2  4304  iinexg  4561  dfiin3g  5094  iinfi  7949  mreiincl  15580  iinopn  20009  clsval2  20142  alexsublem  21137
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