Theorem eliind2 44284

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 412	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐵 → 𝐴 ∈ 𝐶))
4	1, 3	ralrimi 3253	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶)
5		eliind2.2	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
6		eliin 5002	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶))
7	5, 6	syl 17	. 2 ⊢ (𝜑 → (𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶))
8	4, 7	mpbird 257	1 ⊢ (𝜑 → 𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395  Ⅎwnf 1784   ∈ wcel 2105  ∀wral 3060  ∩ ciin 4998

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-12 2170  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1543  df-ex 1781  df-nf 1785  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-iin 5000

This theorem is referenced by:  fsupdm  46020  finfdm  46024