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Theorem dfiin2g 4000
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 2619 . . . 4
2 df-ral 2619 . . . . . 6
3 eleq2 2414 . . . . . . . . . . . . 13
43biimprcd 216 . . . . . . . . . . . 12
54alrimiv 1631 . . . . . . . . . . 11
6 eqid 2353 . . . . . . . . . . . 12
7 eqeq1 2359 . . . . . . . . . . . . . 14
87, 3imbi12d 311 . . . . . . . . . . . . 13
98spcgv 2939 . . . . . . . . . . . 12
106, 9mpii 39 . . . . . . . . . . 11
115, 10impbid2 195 . . . . . . . . . 10
1211imim2i 13 . . . . . . . . 9
1312pm5.74d 238 . . . . . . . 8
1413alimi 1559 . . . . . . 7
15 albi 1564 . . . . . . 7
1614, 15syl 15 . . . . . 6
172, 16sylbi 187 . . . . 5
18 df-ral 2619 . . . . . . . 8
1918albii 1566 . . . . . . 7
20 alcom 1737 . . . . . . 7
2119, 20bitr4i 243 . . . . . 6
22 r19.23v 2730 . . . . . . . 8
23 vex 2862 . . . . . . . . . 10
24 eqeq1 2359 . . . . . . . . . . 11
2524rexbidv 2635 . . . . . . . . . 10
2623, 25elab 2985 . . . . . . . . 9
2726imbi1i 315 . . . . . . . 8
2822, 27bitr4i 243 . . . . . . 7
2928albii 1566 . . . . . 6
30 19.21v 1890 . . . . . . 7
3130albii 1566 . . . . . 6
3221, 29, 313bitr3ri 267 . . . . 5
3317, 32syl6bb 252 . . . 4
341, 33syl5bb 248 . . 3
3534abbidv 2467 . 2
36 df-iin 3972 . 2
37 df-int 3927 . 2
3835, 36, 373eqtr4g 2410 1
Colors of variables: wff setvar class
Syntax hints:   wi 4   wb 176  wal 1540   wceq 1642   wcel 1710  cab 2339  wral 2614  wrex 2615  cint 3926  ciin 3970
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619  df-rex 2620  df-v 2861  df-int 3927  df-iin 3972
This theorem is referenced by:  dfiin2  4002
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