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Theorem iunxiun 3954
Description: Separate an indexed union in the index of an indexed union. (Contributed by Mario Carneiro, 5-Dec-2016.)
Assertion
Ref Expression
iunxiun 𝑥 𝑦𝐴 𝐵𝐶 = 𝑦𝐴 𝑥𝐵 𝐶
Distinct variable groups:   𝑥,𝑦   𝑥,𝐴   𝑦,𝐶
Allowed substitution hints:   𝐴(𝑦)   𝐵(𝑥,𝑦)   𝐶(𝑥)

Proof of Theorem iunxiun
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 eliun 3877 . . . . . . . 8 (𝑥 𝑦𝐴 𝐵 ↔ ∃𝑦𝐴 𝑥𝐵)
21anbi1i 455 . . . . . . 7 ((𝑥 𝑦𝐴 𝐵𝑧𝐶) ↔ (∃𝑦𝐴 𝑥𝐵𝑧𝐶))
3 r19.41v 2626 . . . . . . 7 (∃𝑦𝐴 (𝑥𝐵𝑧𝐶) ↔ (∃𝑦𝐴 𝑥𝐵𝑧𝐶))
42, 3bitr4i 186 . . . . . 6 ((𝑥 𝑦𝐴 𝐵𝑧𝐶) ↔ ∃𝑦𝐴 (𝑥𝐵𝑧𝐶))
54exbii 1598 . . . . 5 (∃𝑥(𝑥 𝑦𝐴 𝐵𝑧𝐶) ↔ ∃𝑥𝑦𝐴 (𝑥𝐵𝑧𝐶))
6 rexcom4 2753 . . . . 5 (∃𝑦𝐴𝑥(𝑥𝐵𝑧𝐶) ↔ ∃𝑥𝑦𝐴 (𝑥𝐵𝑧𝐶))
75, 6bitr4i 186 . . . 4 (∃𝑥(𝑥 𝑦𝐴 𝐵𝑧𝐶) ↔ ∃𝑦𝐴𝑥(𝑥𝐵𝑧𝐶))
8 df-rex 2454 . . . 4 (∃𝑥 𝑦𝐴 𝐵𝑧𝐶 ↔ ∃𝑥(𝑥 𝑦𝐴 𝐵𝑧𝐶))
9 eliun 3877 . . . . . 6 (𝑧 𝑥𝐵 𝐶 ↔ ∃𝑥𝐵 𝑧𝐶)
10 df-rex 2454 . . . . . 6 (∃𝑥𝐵 𝑧𝐶 ↔ ∃𝑥(𝑥𝐵𝑧𝐶))
119, 10bitri 183 . . . . 5 (𝑧 𝑥𝐵 𝐶 ↔ ∃𝑥(𝑥𝐵𝑧𝐶))
1211rexbii 2477 . . . 4 (∃𝑦𝐴 𝑧 𝑥𝐵 𝐶 ↔ ∃𝑦𝐴𝑥(𝑥𝐵𝑧𝐶))
137, 8, 123bitr4i 211 . . 3 (∃𝑥 𝑦𝐴 𝐵𝑧𝐶 ↔ ∃𝑦𝐴 𝑧 𝑥𝐵 𝐶)
14 eliun 3877 . . 3 (𝑧 𝑥 𝑦𝐴 𝐵𝐶 ↔ ∃𝑥 𝑦𝐴 𝐵𝑧𝐶)
15 eliun 3877 . . 3 (𝑧 𝑦𝐴 𝑥𝐵 𝐶 ↔ ∃𝑦𝐴 𝑧 𝑥𝐵 𝐶)
1613, 14, 153bitr4i 211 . 2 (𝑧 𝑥 𝑦𝐴 𝐵𝐶𝑧 𝑦𝐴 𝑥𝐵 𝐶)
1716eqriv 2167 1 𝑥 𝑦𝐴 𝐵𝐶 = 𝑦𝐴 𝑥𝐵 𝐶
Colors of variables: wff set class
Syntax hints:  wa 103   = wceq 1348  wex 1485  wcel 2141  wrex 2449   ciun 3873
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-iun 3875
This theorem is referenced by: (None)
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