Theorem eliind2 45739

Description: Membership in indexed intersection. (Contributed by Glauco Siliprandi, 23-Oct-2021.)

Hypotheses

Ref	Expression
eliind2.1	⊢ Ⅎ𝑥𝜑
eliind2.2	⊢ (𝜑 → 𝐴 ∈ 𝑉)
eliind2.3	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐴 ∈ 𝐶)

Assertion

Ref	Expression
eliind2	⊢ (𝜑 → 𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶)

Distinct variable group:   𝑥,𝐴

Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)   𝐶(𝑥)   𝑉(𝑥)

Proof of Theorem eliind2

Step	Hyp	Ref	Expression
1		eliind2.1	. . 3 ⊢ Ⅎ𝑥𝜑
2		eliind2.3	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐴 ∈ 𝐶)
3	2	ex 417	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐵 → 𝐴 ∈ 𝐶))
4	1, 3	ralrimi 3269	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶)
5		eliind2.2	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
6		eliin 4965	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶))
7	5, 6	syl 18	. 2 ⊢ (𝜑 → (𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶))
8	4, 7	mpbird 260	1 ⊢ (𝜑 → 𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400  Ⅎwnf 1810   ∈ wcel 2149  ∀wral 3085  ∩ ciin 4961

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-12 2219  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1570  df-ex 1807  df-nf 1811  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-iin 4963

This theorem is referenced by:  fsupdm  47447  finfdm  47451