Theorem eliin2 41402

Description: Membership in indexed intersection. See eliincex 41396 for a counterexample showing that the precondition 𝐵 ≠ ∅ cannot be simply dropped. eliin 4924 uses an alternative precondition (and it doesn't have a disjoint var constraint between 𝐵 and 𝑥; see eliin2f 41390). (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 2977	. 2 ⊢ Ⅎ𝑥𝐵
2	1	eliin2f 41390	1 ⊢ (𝐵 ≠ ∅ → (𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶))

Syntax hints:   → wi 4   ↔ wb 208   ∈ wcel 2114   ≠ wne 3016  ∀wral 3138  ∅c0 4291  ∩ ciin 4920

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-nul 4292  df-iin 4922

This theorem is referenced by:  eliuniin2  41406  allbutfi  41685