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Theorem uniiunlem 3353
 Description: A subset relationship useful for converting union to indexed union using dfiun2 4001 or dfiun2g 3999 and intersection to indexed intersection using dfiin2 4002. (Contributed by NM, 5-Oct-2006.) (Proof shortened by Mario Carneiro, 26-Sep-2015.)
Assertion
Ref Expression
uniiunlem (x A B D → (x A B C ↔ {y x A y = B} C))
Distinct variable groups:   x,y   y,A   y,B   x,C
Allowed substitution hints:   A(x)   B(x)   C(y)   D(x,y)

Proof of Theorem uniiunlem
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 eqeq1 2359 . . . . . 6 (y = z → (y = Bz = B))
21rexbidv 2635 . . . . 5 (y = z → (x A y = Bx A z = B))
32cbvabv 2472 . . . 4 {y x A y = B} = {z x A z = B}
43sseq1i 3295 . . 3 ({y x A y = B} C ↔ {z x A z = B} C)
5 r19.23v 2730 . . . . 5 (x A (z = Bz C) ↔ (x A z = Bz C))
65albii 1566 . . . 4 (zx A (z = Bz C) ↔ z(x A z = Bz C))
7 ralcom4 2877 . . . 4 (x A z(z = Bz C) ↔ zx A (z = Bz C))
8 abss 3335 . . . 4 ({z x A z = B} Cz(x A z = Bz C))
96, 7, 83bitr4i 268 . . 3 (x A z(z = Bz C) ↔ {z x A z = B} C)
104, 9bitr4i 243 . 2 ({y x A y = B} Cx A z(z = Bz C))
11 nfv 1619 . . . . 5 z B C
12 eleq1 2413 . . . . 5 (z = B → (z CB C))
1311, 12ceqsalg 2883 . . . 4 (B D → (z(z = Bz C) ↔ B C))
1413ralimi 2689 . . 3 (x A B Dx A (z(z = Bz C) ↔ B C))
15 ralbi 2750 . . 3 (x A (z(z = Bz C) ↔ B C) → (x A z(z = Bz C) ↔ x A B C))
1614, 15syl 15 . 2 (x A B D → (x A z(z = Bz C) ↔ x A B C))
1710, 16syl5rbb 249 1 (x A B D → (x A B C ↔ {y x A y = B} C))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540   = wceq 1642   ∈ wcel 1710  {cab 2339  ∀wral 2614  ∃wrex 2615   ⊆ wss 3257 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-nan 1288  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-nin 3211  df-compl 3212  df-in 3213  df-ss 3259 This theorem is referenced by: (None)
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