Theorem eliin2 45570

Description: Membership in indexed intersection. See eliincex 45564 for a counterexample showing that the precondition 𝐵 ≠ ∅ cannot be simply dropped. eliin 4933 uses an alternative precondition (and it doesn't have a disjoint var constraint between 𝐵 and 𝑥; see eliin2f 45558). (Contributed by Glauco Siliprandi, 26-Jun-2021.)

Assertion

Ref	Expression
eliin2	⊢ (𝐵 ≠ ∅ → (𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Allowed substitution hint:   𝐶(𝑥)

Proof of Theorem eliin2

Step	Hyp	Ref	Expression
1		nfcv 2902	. 2 ⊢ Ⅎ𝑥𝐵
2	1	eliin2f 45558	1 ⊢ (𝐵 ≠ ∅ → (𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶))

Syntax hints:   → wi 4   ↔ wb 207   ∈ wcel 2119   ≠ wne 2935  ∀wral 3054  ∅c0 4268  ∩ ciin 4929

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-nul 4269  df-iin 4931

This theorem is referenced by:  eliuniin2  45574  allbutfi  45844