Theorem eliin2 45086

Description: Membership in indexed intersection. See eliincex 45080 for a counterexample showing that the precondition 𝐵 ≠ ∅ cannot be simply dropped. eliin 5004 uses an alternative precondition (and it doesn't have a disjoint var constraint between 𝐵 and 𝑥; see eliin2f 45074). (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 2905	. 2 ⊢ Ⅎ𝑥𝐵
2	1	eliin2f 45074	1 ⊢ (𝐵 ≠ ∅ → (𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶))

Syntax hints:   → wi 4   ↔ wb 206   ∈ wcel 2108   ≠ wne 2940  ∀wral 3061  ∅c0 4342  ∩ ciin 5000

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-v 3483  df-sbc 3795  df-csb 3912  df-dif 3969  df-nul 4343  df-iin 5002

This theorem is referenced by:  eliuniin2  45090  allbutfi  45372