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Theorem eliinid 43226
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 484 . . 3 ((𝐴 𝑥𝐵 𝐶𝑥𝐵) → 𝐴 𝑥𝐵 𝐶)
2 eliin 4958 . . . 4 (𝐴 𝑥𝐵 𝐶 → (𝐴 𝑥𝐵 𝐶 ↔ ∀𝑥𝐵 𝐴𝐶))
32adantr 482 . . 3 ((𝐴 𝑥𝐵 𝐶𝑥𝐵) → (𝐴 𝑥𝐵 𝐶 ↔ ∀𝑥𝐵 𝐴𝐶))
41, 3mpbid 231 . 2 ((𝐴 𝑥𝐵 𝐶𝑥𝐵) → ∀𝑥𝐵 𝐴𝐶)
5 rspa 3230 . 2 ((∀𝑥𝐵 𝐴𝐶𝑥𝐵) → 𝐴𝐶)
64, 5sylancom 589 1 ((𝐴 𝑥𝐵 𝐶𝑥𝐵) → 𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  wcel 2107  wral 3063   ciin 4954
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-12 2172  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3064  df-iin 4956
This theorem is referenced by:  iinssiin  43244  fnlimfvre  43810  smflimlem2  44908  smflimmpt  44946  smfsuplem1  44947  smfsupmpt  44951  smfsupxr  44952  smfinflem  44953  smfinfmpt  44955  smflimsuplem4  44959  smflimsupmpt  44965  smfliminfmpt  44968  fsupdm  44978  finfdm  44982
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