Theorem eliin2 45152

Description: Membership in indexed intersection. See eliincex 45146 for a counterexample showing that the precondition 𝐵 ≠ ∅ cannot be simply dropped. eliin 4946 uses an alternative precondition (and it doesn't have a disjoint var constraint between 𝐵 and 𝑥; see eliin2f 45140). (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 2894	. 2 ⊢ Ⅎ𝑥𝐵
2	1	eliin2f 45140	1 ⊢ (𝐵 ≠ ∅ → (𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶))

Syntax hints:   → wi 4   ↔ wb 206   ∈ wcel 2111   ≠ wne 2928  ∀wral 3047  ∅c0 4283  ∩ ciin 4942

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-nul 4284  df-iin 4944

This theorem is referenced by:  eliuniin2  45156  allbutfi  45430