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Theorem ssiun3 29683
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 3732 . 2 (𝐶 𝑥𝐴 𝐵 ↔ ∀𝑦(𝑦𝐶𝑦 𝑥𝐴 𝐵))
2 df-ral 3055 . 2 (∀𝑦𝐶 𝑦 𝑥𝐴 𝐵 ↔ ∀𝑦(𝑦𝐶𝑦 𝑥𝐴 𝐵))
3 eliun 4676 . . 3 (𝑦 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 𝑦𝐵)
43ralbii 3118 . 2 (∀𝑦𝐶 𝑦 𝑥𝐴 𝐵 ↔ ∀𝑦𝐶𝑥𝐴 𝑦𝐵)
51, 2, 43bitr2ri 289 1 (∀𝑦𝐶𝑥𝐴 𝑦𝐵𝐶 𝑥𝐴 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1630  wcel 2139  wral 3050  wrex 3051  wss 3715   ciun 4672
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-v 3342  df-in 3722  df-ss 3729  df-iun 4674
This theorem is referenced by: (None)
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