Theorem iinab 5010

Description: Indexed intersection of a class abstraction. (Contributed by NM, 6-Dec-2011.)

Assertion

Ref	Expression
iinab	⊢ ∩ 𝑥 ∈ 𝐴 {𝑦 ∣ 𝜑} = {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝜑}

Distinct variable groups:   𝑦,𝐴   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥)

Proof of Theorem iinab

Step	Hyp	Ref	Expression
1		nfcv 2898	. . 3 ⊢ Ⅎ𝑦𝐴
2		nfab1 2900	. . 3 ⊢ Ⅎ𝑦{𝑦 ∣ 𝜑}
3	1, 2	nfiin 4966	. 2 ⊢ Ⅎ𝑦∩ 𝑥 ∈ 𝐴 {𝑦 ∣ 𝜑}
4		nfab1 2900	. 2 ⊢ Ⅎ𝑦{𝑦 ∣ ∀𝑥 ∈ 𝐴 𝜑}
5		abid 2718	. . . 4 ⊢ (𝑦 ∈ {𝑦 ∣ 𝜑} ↔ 𝜑)
6	5	ralbii 3083	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝑦 ∈ {𝑦 ∣ 𝜑} ↔ ∀𝑥 ∈ 𝐴 𝜑)
7		eliin 4938	. . . 4 ⊢ (𝑦 ∈ V → (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 {𝑦 ∣ 𝜑} ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ {𝑦 ∣ 𝜑}))
8	7	elv 3434	. . 3 ⊢ (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 {𝑦 ∣ 𝜑} ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ {𝑦 ∣ 𝜑})
9		abid 2718	. . 3 ⊢ (𝑦 ∈ {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝜑} ↔ ∀𝑥 ∈ 𝐴 𝜑)
10	6, 8, 9	3bitr4i 303	. 2 ⊢ (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 {𝑦 ∣ 𝜑} ↔ 𝑦 ∈ {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝜑})
11	3, 4, 10	eqri 3942	1 ⊢ ∩ 𝑥 ∈ 𝐴 {𝑦 ∣ 𝜑} = {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝜑}

Syntax hints:   ↔ wb 206   = wceq 1542   ∈ wcel 2114  {cab 2714  ∀wral 3051  Vcvv 3429  ∩ ciin 4934

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ral 3052  df-v 3431  df-iin 4936

This theorem is referenced by:  iinrab  5011