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Theorem ssiun3 32571
Description: Subset equivalence for an indexed union. (Contributed by Thierry Arnoux, 17-Oct-2016.)
Assertion
Ref Expression
ssiun3 (∀𝑦𝐶𝑥𝐴 𝑦𝐵𝐶 𝑥𝐴 𝐵)
Distinct variable groups:   𝑥,𝑦   𝑦,𝐴   𝑦,𝐵   𝑦,𝐶
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝐶(𝑥)

Proof of Theorem ssiun3
StepHypRef Expression
1 df-ss 3968 . 2 (𝐶 𝑥𝐴 𝐵 ↔ ∀𝑦(𝑦𝐶𝑦 𝑥𝐴 𝐵))
2 df-ral 3062 . 2 (∀𝑦𝐶 𝑦 𝑥𝐴 𝐵 ↔ ∀𝑦(𝑦𝐶𝑦 𝑥𝐴 𝐵))
3 eliun 4995 . . 3 (𝑦 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 𝑦𝐵)
43ralbii 3093 . 2 (∀𝑦𝐶 𝑦 𝑥𝐴 𝐵 ↔ ∀𝑦𝐶𝑥𝐴 𝑦𝐵)
51, 2, 43bitr2ri 300 1 (∀𝑦𝐶𝑥𝐴 𝑦𝐵𝐶 𝑥𝐴 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wal 1538  wcel 2108  wral 3061  wrex 3070  wss 3951   ciun 4991
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-v 3482  df-ss 3968  df-iun 4993
This theorem is referenced by: (None)
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