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Theorem dfiunv2 3684
 Description: Define double indexed union. (Contributed by FL, 6-Nov-2013.)
Assertion
Ref Expression
dfiunv2 x A y B 𝐶 = {zx A y B z 𝐶}
Distinct variable groups:   x,z   y,z   z,A   z,B   z,𝐶
Allowed substitution hints:   A(x,y)   B(x,y)   𝐶(x,y)

Proof of Theorem dfiunv2
Dummy variable w is distinct from all other variables.
StepHypRef Expression
1 df-iun 3650 . . . 4 y B 𝐶 = {wy B w 𝐶}
21a1i 9 . . 3 (x A y B 𝐶 = {wy B w 𝐶})
32iuneq2i 3666 . 2 x A y B 𝐶 = x A {wy B w 𝐶}
4 df-iun 3650 . 2 x A {wy B w 𝐶} = {zx A z {wy B w 𝐶}}
5 vex 2554 . . . . 5 z V
6 eleq1 2097 . . . . . 6 (w = z → (w 𝐶z 𝐶))
76rexbidv 2321 . . . . 5 (w = z → (y B w 𝐶y B z 𝐶))
85, 7elab 2681 . . . 4 (z {wy B w 𝐶} ↔ y B z 𝐶)
98rexbii 2325 . . 3 (x A z {wy B w 𝐶} ↔ x A y B z 𝐶)
109abbii 2150 . 2 {zx A z {wy B w 𝐶}} = {zx A y B z 𝐶}
113, 4, 103eqtri 2061 1 x A y B 𝐶 = {zx A y B z 𝐶}
 Colors of variables: wff set class Syntax hints:   = 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-in 2918  df-ss 2925  df-iun 3650 This theorem is referenced by: (None)
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