Theorem inrab2 3400

Description: Intersection with a restricted class abstraction. (Contributed by NM, 19-Nov-2007.)

Assertion

Ref	Expression
inrab2	⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∩ 𝐵) = {𝑥 ∈ (𝐴 ∩ 𝐵) ∣ 𝜑}

Distinct variable group:   𝑥,𝐵

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

Proof of Theorem inrab2

Step	Hyp	Ref	Expression
1		df-rab 2457	. . 3 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
2		abid2 2291	. . . 4 ⊢ {𝑥 ∣ 𝑥 ∈ 𝐵} = 𝐵
3	2	eqcomi 2174	. . 3 ⊢ 𝐵 = {𝑥 ∣ 𝑥 ∈ 𝐵}
4	1, 3	ineq12i 3326	. 2 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∩ 𝐵) = ({𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ∩ {𝑥 ∣ 𝑥 ∈ 𝐵})
5		df-rab 2457	. . 3 ⊢ {𝑥 ∈ (𝐴 ∩ 𝐵) ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ (𝐴 ∩ 𝐵) ∧ 𝜑)}
6		inab 3395	. . . 4 ⊢ ({𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ∩ {𝑥 ∣ 𝑥 ∈ 𝐵}) = {𝑥 ∣ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ 𝑥 ∈ 𝐵)}
7		elin 3310	. . . . . . 7 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))
8	7	anbi1i 455	. . . . . 6 ⊢ ((𝑥 ∈ (𝐴 ∩ 𝐵) ∧ 𝜑) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∧ 𝜑))
9		an32 557	. . . . . 6 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∧ 𝜑) ↔ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ 𝑥 ∈ 𝐵))
10	8, 9	bitri 183	. . . . 5 ⊢ ((𝑥 ∈ (𝐴 ∩ 𝐵) ∧ 𝜑) ↔ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ 𝑥 ∈ 𝐵))
11	10	abbii 2286	. . . 4 ⊢ {𝑥 ∣ (𝑥 ∈ (𝐴 ∩ 𝐵) ∧ 𝜑)} = {𝑥 ∣ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ 𝑥 ∈ 𝐵)}
12	6, 11	eqtr4i 2194	. . 3 ⊢ ({𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ∩ {𝑥 ∣ 𝑥 ∈ 𝐵}) = {𝑥 ∣ (𝑥 ∈ (𝐴 ∩ 𝐵) ∧ 𝜑)}
13	5, 12	eqtr4i 2194	. 2 ⊢ {𝑥 ∈ (𝐴 ∩ 𝐵) ∣ 𝜑} = ({𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ∩ {𝑥 ∣ 𝑥 ∈ 𝐵})
14	4, 13	eqtr4i 2194	1 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∩ 𝐵) = {𝑥 ∈ (𝐴 ∩ 𝐵) ∣ 𝜑}

Syntax hints:   ∧ wa 103   = wceq 1348   ∈ wcel 2141  {cab 2156  {crab 2452   ∩ cin 3120

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rab 2457  df-v 2732  df-in 3127

This theorem is referenced by:  iooval2  9872  fzval2  9968  dfphi2  12174  znnen  12353