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Theorem eliinid 39762
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 474 . . 3 ((𝐴 𝑥𝐵 𝐶𝑥𝐵) → 𝐴 𝑥𝐵 𝐶)
2 eliin 4665 . . . 4 (𝐴 𝑥𝐵 𝐶 → (𝐴 𝑥𝐵 𝐶 ↔ ∀𝑥𝐵 𝐴𝐶))
32adantr 472 . . 3 ((𝐴 𝑥𝐵 𝐶𝑥𝐵) → (𝐴 𝑥𝐵 𝐶 ↔ ∀𝑥𝐵 𝐴𝐶))
41, 3mpbid 222 . 2 ((𝐴 𝑥𝐵 𝐶𝑥𝐵) → ∀𝑥𝐵 𝐴𝐶)
5 rspa 3056 . 2 ((∀𝑥𝐵 𝐴𝐶𝑥𝐵) → 𝐴𝐶)
64, 5sylancom 704 1 ((𝐴 𝑥𝐵 𝐶𝑥𝐵) → 𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wcel 2127  wral 3038   ciin 4661
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ral 3043  df-v 3330  df-iin 4663
This theorem is referenced by:  iinssiin  39780  fnlimfvre  40378  smflimlem2  41455  smflimmpt  41491  smfsuplem1  41492  smfsupmpt  41496  smfsupxr  41497  smfinflem  41498  smfinfmpt  41500  smflimsuplem4  41504  smflimsupmpt  41510  smfliminfmpt  41513
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