Theorem iinab 5030

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 2924	. . 3 ⊢ Ⅎ𝑦𝐴
2		nfab1 2926	. . 3 ⊢ Ⅎ𝑦{𝑦 ∣ 𝜑}
3	1, 2	nfiin 4987	. 2 ⊢ Ⅎ𝑦∩ 𝑥 ∈ 𝐴 {𝑦 ∣ 𝜑}
4		nfab1 2926	. 2 ⊢ Ⅎ𝑦{𝑦 ∣ ∀𝑥 ∈ 𝐴 𝜑}
5		abid 2744	. . . 4 ⊢ (𝑦 ∈ {𝑦 ∣ 𝜑} ↔ 𝜑)
6	5	ralbii 3110	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝑦 ∈ {𝑦 ∣ 𝜑} ↔ ∀𝑥 ∈ 𝐴 𝜑)
7		eliin 4959	. . . 4 ⊢ (𝑦 ∈ V → (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 {𝑦 ∣ 𝜑} ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ {𝑦 ∣ 𝜑}))
8	7	elv 3458	. . 3 ⊢ (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 {𝑦 ∣ 𝜑} ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ {𝑦 ∣ 𝜑})
9		abid 2744	. . 3 ⊢ (𝑦 ∈ {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝜑} ↔ ∀𝑥 ∈ 𝐴 𝜑)
10	6, 8, 9	3bitr4i 306	. 2 ⊢ (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 {𝑦 ∣ 𝜑} ↔ 𝑦 ∈ {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝜑})
11	3, 4, 10	eqri 3954	1 ⊢ ∩ 𝑥 ∈ 𝐴 {𝑦 ∣ 𝜑} = {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝜑}

This proof depends on syntax axioms:   ↔ wb 209   = wceq 1570   ∈ wcel 2145  {cab 2740  ∀wral 3078  Vcvv 3453  ∩ ciin 4955

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ral 3079  df-v 3455  df-iin 4957

This theorem is used by:  iinrab  5031