Theorem dfrab2 3378

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 2441	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
2		inab 3371	. . 3 ⊢ ({𝑥 ∣ 𝑥 ∈ 𝐴} ∩ {𝑥 ∣ 𝜑}) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
3		abid2 2275	. . . 4 ⊢ {𝑥 ∣ 𝑥 ∈ 𝐴} = 𝐴
4	3	ineq1i 3300	. . 3 ⊢ ({𝑥 ∣ 𝑥 ∈ 𝐴} ∩ {𝑥 ∣ 𝜑}) = (𝐴 ∩ {𝑥 ∣ 𝜑})
5	2, 4	eqtr3i 2177	. 2 ⊢ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} = (𝐴 ∩ {𝑥 ∣ 𝜑})
6		incom 3295	. 2 ⊢ (𝐴 ∩ {𝑥 ∣ 𝜑}) = ({𝑥 ∣ 𝜑} ∩ 𝐴)
7	1, 5, 6	3eqtri 2179	1 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = ({𝑥 ∣ 𝜑} ∩ 𝐴)

Syntax hints:   ∧ wa 103   = wceq 1332   ∈ wcel 2125  {cab 2140  {crab 2436   ∩ cin 3097

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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-ext 2136

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1740  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-rab 2441  df-v 2711  df-in 3104

This theorem is referenced by:  minmax  11106  xrminmax  11139