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Theorem ssiun3 32367
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 dfss2 3967 . 2 (𝐶 𝑥𝐴 𝐵 ↔ ∀𝑦(𝑦𝐶𝑦 𝑥𝐴 𝐵))
2 df-ral 3058 . 2 (∀𝑦𝐶 𝑦 𝑥𝐴 𝐵 ↔ ∀𝑦(𝑦𝐶𝑦 𝑥𝐴 𝐵))
3 eliun 5002 . . 3 (𝑦 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 𝑦𝐵)
43ralbii 3089 . 2 (∀𝑦𝐶 𝑦 𝑥𝐴 𝐵 ↔ ∀𝑦𝐶𝑥𝐴 𝑦𝐵)
51, 2, 43bitr2ri 299 1 (∀𝑦𝐶𝑥𝐴 𝑦𝐵𝐶 𝑥𝐴 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wal 1531  wcel 2098  wral 3057  wrex 3066  wss 3947   ciun 4998
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2698
This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2705  df-cleq 2719  df-clel 2805  df-ral 3058  df-rex 3067  df-v 3473  df-in 3954  df-ss 3964  df-iun 5000
This theorem is referenced by: (None)
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