Theorem eliin2 45814

Description: Membership in indexed intersection. See eliincex 45808 for a counterexample showing that the precondition 𝐵 ≠ ∅ cannot be simply dropped. eliin 4962 uses an alternative precondition (and it doesn't have a disjoint var constraint between 𝐵 and 𝑥; see eliin2f 45802). (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 2925	. 2 ⊢ Ⅎ𝑥𝐵
2	1	eliin2f 45802	1 ⊢ (𝐵 ≠ ∅ → (𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶))

Syntax hints:   → wi 4   ↔ wb 209   ∈ wcel 2143   ≠ wne 2958  ∀wral 3079  ∅c0 4287  ∩ ciin 4958

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-nul 4288  df-iin 4960

This theorem is referenced by:  eliuniin2  45818  allbutfi  46088