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Theorem iunab 3694
Description: The indexed union of a class abstraction. (Contributed by NM, 27-Dec-2004.)
Assertion
Ref Expression
iunab x A {yφ} = {yx 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 2175 . . . 4 yA
2 nfab1 2177 . . . 4 y{yφ}
31, 2nfiunxy 3674 . . 3 y x A {yφ}
4 nfab1 2177 . . 3 y{yx A φ}
53, 4cleqf 2198 . 2 ( x A {yφ} = {yx A φ} ↔ y(y x A {yφ} ↔ y {yx A φ}))
6 abid 2025 . . . 4 (y {yφ} ↔ φ)
76rexbii 2325 . . 3 (x A y {yφ} ↔ x A φ)
8 eliun 3652 . . 3 (y x A {yφ} ↔ x A y {yφ})
9 abid 2025 . . 3 (y {yx A φ} ↔ x A φ)
107, 8, 93bitr4i 201 . 2 (y x A {yφ} ↔ y {yx A φ})
115, 10mpgbir 1339 1 x A {yφ} = {yx A φ}
Colors of variables: wff set class
Syntax hints:  wb 98   = wceq 1242   wcel 1390  {cab 2023  wrex 2301   ciun 3648
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-iun 3650
This theorem is referenced by:  iunrab  3695  iunid  3703  dfimafn2  5166
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