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Theorem dfiin2g 4000
 Description: Alternate definition of indexed intersection when B is a set. (Contributed by Jeff Hankins, 27-Aug-2009.)
Assertion
Ref Expression
dfiin2g (x A B Cx A B = {y x A y = B})
Distinct variable groups:   y,A   y,B   x,y
Allowed substitution hints:   A(x)   B(x)   C(x,y)

Proof of Theorem dfiin2g
Dummy variables z w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ral 2619 . . . 4 (x A w Bx(x Aw B))
2 df-ral 2619 . . . . . 6 (x A B Cx(x AB C))
3 eleq2 2414 . . . . . . . . . . . . 13 (z = B → (w zw B))
43biimprcd 216 . . . . . . . . . . . 12 (w B → (z = Bw z))
54alrimiv 1631 . . . . . . . . . . 11 (w Bz(z = Bw z))
6 eqid 2353 . . . . . . . . . . . 12 B = B
7 eqeq1 2359 . . . . . . . . . . . . . 14 (z = B → (z = BB = B))
87, 3imbi12d 311 . . . . . . . . . . . . 13 (z = B → ((z = Bw z) ↔ (B = Bw B)))
98spcgv 2939 . . . . . . . . . . . 12 (B C → (z(z = Bw z) → (B = Bw B)))
106, 9mpii 39 . . . . . . . . . . 11 (B C → (z(z = Bw z) → w B))
115, 10impbid2 195 . . . . . . . . . 10 (B C → (w Bz(z = Bw z)))
1211imim2i 13 . . . . . . . . 9 ((x AB C) → (x A → (w Bz(z = Bw z))))
1312pm5.74d 238 . . . . . . . 8 ((x AB C) → ((x Aw B) ↔ (x Az(z = Bw z))))
1413alimi 1559 . . . . . . 7 (x(x AB C) → x((x Aw B) ↔ (x Az(z = Bw z))))
15 albi 1564 . . . . . . 7 (x((x Aw B) ↔ (x Az(z = Bw z))) → (x(x Aw B) ↔ x(x Az(z = Bw z))))
1614, 15syl 15 . . . . . 6 (x(x AB C) → (x(x Aw B) ↔ x(x Az(z = Bw z))))
172, 16sylbi 187 . . . . 5 (x A B C → (x(x Aw B) ↔ x(x Az(z = Bw z))))
18 df-ral 2619 . . . . . . . 8 (x A (z = Bw z) ↔ x(x A → (z = Bw z)))
1918albii 1566 . . . . . . 7 (zx A (z = Bw z) ↔ zx(x A → (z = Bw z)))
20 alcom 1737 . . . . . . 7 (xz(x A → (z = Bw z)) ↔ zx(x A → (z = Bw z)))
2119, 20bitr4i 243 . . . . . 6 (zx A (z = Bw z) ↔ xz(x A → (z = Bw z)))
22 r19.23v 2730 . . . . . . . 8 (x A (z = Bw z) ↔ (x A z = Bw z))
23 vex 2862 . . . . . . . . . 10 z V
24 eqeq1 2359 . . . . . . . . . . 11 (y = z → (y = Bz = B))
2524rexbidv 2635 . . . . . . . . . 10 (y = z → (x A y = Bx A z = B))
2623, 25elab 2985 . . . . . . . . 9 (z {y x A y = B} ↔ x A z = B)
2726imbi1i 315 . . . . . . . 8 ((z {y x A y = B} → w z) ↔ (x A z = Bw z))
2822, 27bitr4i 243 . . . . . . 7 (x A (z = Bw z) ↔ (z {y x A y = B} → w z))
2928albii 1566 . . . . . 6 (zx A (z = Bw z) ↔ z(z {y x A y = B} → w z))
30 19.21v 1890 . . . . . . 7 (z(x A → (z = Bw z)) ↔ (x Az(z = Bw z)))
3130albii 1566 . . . . . 6 (xz(x A → (z = Bw z)) ↔ x(x Az(z = Bw z)))
3221, 29, 313bitr3ri 267 . . . . 5 (x(x Az(z = Bw z)) ↔ z(z {y x A y = B} → w z))
3317, 32syl6bb 252 . . . 4 (x A B C → (x(x Aw B) ↔ z(z {y x A y = B} → w z)))
341, 33syl5bb 248 . . 3 (x A B C → (x A w Bz(z {y x A y = B} → w z)))
3534abbidv 2467 . 2 (x A B C → {w x A w B} = {w z(z {y x A y = B} → w z)})
36 df-iin 3972 . 2 x A B = {w x A w B}
37 df-int 3927 . 2 {y x A y = B} = {w z(z {y x A y = B} → w z)}
3835, 36, 373eqtr4g 2410 1 (x A B Cx A B = {y x A y = B})
 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-1 5  ax-2 6  ax-3 7  ax-mp 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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