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Theorem iunab 4012
 Description: The indexed union of a class abstraction. (Contributed by NM, 27-Dec-2004.)
Assertion
Ref Expression
iunab x A {y φ} = {y x A φ}
Distinct variable groups:   y,A   x,y
Allowed substitution hints:   φ(x,y)   A(x)

Proof of Theorem iunab
StepHypRef Expression
1 nfcv 2489 . . . 4 yA
2 nfab1 2491 . . . 4 y{y φ}
31, 2nfiun 3995 . . 3 yx A {y φ}
4 nfab1 2491 . . 3 y{y x A φ}
53, 4cleqf 2513 . 2 (x A {y φ} = {y x A φ} ↔ y(y x A {y φ} ↔ y {y x A φ}))
6 abid 2341 . . . 4 (y {y φ} ↔ φ)
76rexbii 2639 . . 3 (x A y {y φ} ↔ x A φ)
8 eliun 3973 . . 3 (y x A {y φ} ↔ x A y {y φ})
9 abid 2341 . . 3 (y {y x A φ} ↔ x A φ)
107, 8, 93bitr4i 268 . 2 (y x A {y φ} ↔ y {y x A φ})
115, 10mpgbir 1550 1 x A {y φ} = {y x A φ}
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   = wceq 1642   ∈ wcel 1710  {cab 2339  ∃wrex 2615  ∪ciun 3969 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-iun 3971 This theorem is referenced by:  iunrab  4013  iunid  4021  dfimafn2  5367
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