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Theorem eliinid 45116
Description: Membership in an indexed intersection implies membership in any intersected set. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
Assertion
Ref Expression
eliinid ((𝐴 𝑥𝐵 𝐶𝑥𝐵) → 𝐴𝐶)
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑥)

Proof of Theorem eliinid
StepHypRef Expression
1 simpl 482 . . 3 ((𝐴 𝑥𝐵 𝐶𝑥𝐵) → 𝐴 𝑥𝐵 𝐶)
2 eliin 4996 . . . 4 (𝐴 𝑥𝐵 𝐶 → (𝐴 𝑥𝐵 𝐶 ↔ ∀𝑥𝐵 𝐴𝐶))
32adantr 480 . . 3 ((𝐴 𝑥𝐵 𝐶𝑥𝐵) → (𝐴 𝑥𝐵 𝐶 ↔ ∀𝑥𝐵 𝐴𝐶))
41, 3mpbid 232 . 2 ((𝐴 𝑥𝐵 𝐶𝑥𝐵) → ∀𝑥𝐵 𝐴𝐶)
5 rspa 3248 . 2 ((∀𝑥𝐵 𝐴𝐶𝑥𝐵) → 𝐴𝐶)
64, 5sylancom 588 1 ((𝐴 𝑥𝐵 𝐶𝑥𝐵) → 𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wcel 2108  wral 3061   ciin 4992
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-12 2177  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-iin 4994
This theorem is referenced by:  iinssiin  45134  fnlimfvre  45689  smflimlem2  46787  smflimmpt  46825  smfsuplem1  46826  smfsupmpt  46830  smfsupxr  46831  smfinflem  46832  smfinfmpt  46834  smflimsuplem4  46838  smflimsupmpt  46844  smfliminfmpt  46847  fsupdm  46857  finfdm  46861
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