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Theorem iunss1 3824
 Description: Subclass theorem for indexed union. (Contributed by NM, 10-Dec-2004.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
Assertion
Ref Expression
iunss1 (𝐴𝐵 𝑥𝐴 𝐶 𝑥𝐵 𝐶)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hint:   𝐶(𝑥)

Proof of Theorem iunss1
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 ssrexv 3162 . . 3 (𝐴𝐵 → (∃𝑥𝐴 𝑦𝐶 → ∃𝑥𝐵 𝑦𝐶))
2 eliun 3817 . . 3 (𝑦 𝑥𝐴 𝐶 ↔ ∃𝑥𝐴 𝑦𝐶)
3 eliun 3817 . . 3 (𝑦 𝑥𝐵 𝐶 ↔ ∃𝑥𝐵 𝑦𝐶)
41, 2, 33imtr4g 204 . 2 (𝐴𝐵 → (𝑦 𝑥𝐴 𝐶𝑦 𝑥𝐵 𝐶))
54ssrdv 3103 1 (𝐴𝐵 𝑥𝐴 𝐶 𝑥𝐵 𝐶)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∈ wcel 1480  ∃wrex 2417   ⊆ wss 3071  ∪ ciun 3813 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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121 This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-in 3077  df-ss 3084  df-iun 3815 This theorem is referenced by:  iuneq1  3826  iunxdif2  3861  fsumiun  11246
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