Theorem dfrab2 3484

Description: Alternate definition of restricted class abstraction. (Contributed by NM, 20-Sep-2003.)

Assertion

Ref	Expression
dfrab2	⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = ({𝑥 ∣ 𝜑} ∩ 𝐴)

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem dfrab2

Step	Hyp	Ref	Expression
1		df-rab 2520	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
2		inab 3477	. . 3 ⊢ ({𝑥 ∣ 𝑥 ∈ 𝐴} ∩ {𝑥 ∣ 𝜑}) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
3		abid2 2353	. . . 4 ⊢ {𝑥 ∣ 𝑥 ∈ 𝐴} = 𝐴
4	3	ineq1i 3406	. . 3 ⊢ ({𝑥 ∣ 𝑥 ∈ 𝐴} ∩ {𝑥 ∣ 𝜑}) = (𝐴 ∩ {𝑥 ∣ 𝜑})
5	2, 4	eqtr3i 2254	. 2 ⊢ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} = (𝐴 ∩ {𝑥 ∣ 𝜑})
6		incom 3401	. 2 ⊢ (𝐴 ∩ {𝑥 ∣ 𝜑}) = ({𝑥 ∣ 𝜑} ∩ 𝐴)
7	1, 5, 6	3eqtri 2256	1 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = ({𝑥 ∣ 𝜑} ∩ 𝐴)

Syntax hints:   ∧ wa 104   = wceq 1398   ∈ wcel 2202  {cab 2217  {crab 2515   ∩ cin 3200

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-rab 2520  df-v 2805  df-in 3207

This theorem is referenced by:  minmax  11851  xrminmax  11886