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Theorem ssiun3 30307
 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 3938 . 2 (𝐶 𝑥𝐴 𝐵 ↔ ∀𝑦(𝑦𝐶𝑦 𝑥𝐴 𝐵))
2 df-ral 3137 . 2 (∀𝑦𝐶 𝑦 𝑥𝐴 𝐵 ↔ ∀𝑦(𝑦𝐶𝑦 𝑥𝐴 𝐵))
3 eliun 4904 . . 3 (𝑦 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 𝑦𝐵)
43ralbii 3159 . 2 (∀𝑦𝐶 𝑦 𝑥𝐴 𝐵 ↔ ∀𝑦𝐶𝑥𝐴 𝑦𝐵)
51, 2, 43bitr2ri 303 1 (∀𝑦𝐶𝑥𝐴 𝑦𝐵𝐶 𝑥𝐴 𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1536   ∈ wcel 2115  ∀wral 3132  ∃wrex 3133   ⊆ wss 3918  ∪ ciun 4900 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3137  df-rex 3138  df-v 3481  df-in 3925  df-ss 3935  df-iun 4902 This theorem is referenced by: (None)
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