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Theorem dfiin2g 4126
 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 2712 . . . 4
2 df-ral 2712 . . . . . 6
3 eleq2 2499 . . . . . . . . . . . . 13
43biimprcd 218 . . . . . . . . . . . 12
54alrimiv 1642 . . . . . . . . . . 11
6 eqid 2438 . . . . . . . . . . . 12
7 eqeq1 2444 . . . . . . . . . . . . . 14
87, 3imbi12d 313 . . . . . . . . . . . . 13
98spcgv 3038 . . . . . . . . . . . 12
106, 9mpii 42 . . . . . . . . . . 11
115, 10impbid2 197 . . . . . . . . . 10
1211imim2i 14 . . . . . . . . 9
1312pm5.74d 240 . . . . . . . 8
1413alimi 1569 . . . . . . 7
15 albi 1574 . . . . . . 7
1614, 15syl 16 . . . . . 6
172, 16sylbi 189 . . . . 5
18 df-ral 2712 . . . . . . . 8
1918albii 1576 . . . . . . 7
20 alcom 1753 . . . . . . 7
2119, 20bitr4i 245 . . . . . 6
22 r19.23v 2824 . . . . . . . 8
23 vex 2961 . . . . . . . . . 10
24 eqeq1 2444 . . . . . . . . . . 11
2524rexbidv 2728 . . . . . . . . . 10
2623, 25elab 3084 . . . . . . . . 9
2726imbi1i 317 . . . . . . . 8
2822, 27bitr4i 245 . . . . . . 7
2928albii 1576 . . . . . 6
30 19.21v 1914 . . . . . . 7
3130albii 1576 . . . . . 6
3221, 29, 313bitr3ri 269 . . . . 5
3317, 32syl6bb 254 . . . 4
341, 33syl5bb 250 . . 3
3534abbidv 2552 . 2
36 df-iin 4098 . 2
37 df-int 4053 . 2
3835, 36, 373eqtr4g 2495 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 178  wal 1550   wceq 1653   wcel 1726  cab 2424  wral 2707  wrex 2708  cint 4052  ciin 4096 This theorem is referenced by:  dfiin2  4128  iinexg  4363  dfiin3g  5126  iinfi  7425  mreiincl  13826  iinopn  16980  clsval2  17119  alexsublem  18080 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 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419 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 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rex 2713  df-v 2960  df-int 4053  df-iin 4098
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