Theorem eliind 42619

Description: Membership in indexed intersection. (Contributed by Glauco Siliprandi, 24-Dec-2020.)

Hypotheses

Ref	Expression
eliind.a	⊢ (𝜑 → 𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶)
eliind.k	⊢ (𝜑 → 𝐾 ∈ 𝐵)
eliind.d	⊢ (𝑥 = 𝐾 → (𝐴 ∈ 𝐶 ↔ 𝐴 ∈ 𝐷))

Assertion

Ref	Expression
eliind	⊢ (𝜑 → 𝐴 ∈ 𝐷)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐷   𝑥,𝐾

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

Proof of Theorem eliind

Step	Hyp	Ref	Expression
1		eliind.d	. 2 ⊢ (𝑥 = 𝐾 → (𝐴 ∈ 𝐶 ↔ 𝐴 ∈ 𝐷))
2		eliind.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶)
3		eliin 4929	. . . 4 ⊢ (𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 → (𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶))
4	2, 3	syl 17	. . 3 ⊢ (𝜑 → (𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶))
5	2, 4	mpbid 231	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶)
6		eliind.k	. 2 ⊢ (𝜑 → 𝐾 ∈ 𝐵)
7	1, 5, 6	rspcdva 3562	1 ⊢ (𝜑 → 𝐴 ∈ 𝐷)

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1539   ∈ wcel 2106  ∀wral 3064  ∩ ciin 4925

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-iin 4927

This theorem is referenced by:  iooiinioc  43094  hspdifhsp  44154  smflimlem3  44308  smfsuplem1  44344  smflimsuplem4  44356