Theorem dfrab3 4272

Description: Alternate definition of restricted class abstraction. (Contributed by Mario Carneiro, 8-Sep-2013.)

Assertion

Ref	Expression
dfrab3	⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = (𝐴 ∩ {𝑥 ∣ 𝜑})

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem dfrab3

Step	Hyp	Ref	Expression
1		df-rab 3397	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
2		inab 4262	. 2 ⊢ ({𝑥 ∣ 𝑥 ∈ 𝐴} ∩ {𝑥 ∣ 𝜑}) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
3		abid2 2865	. . 3 ⊢ {𝑥 ∣ 𝑥 ∈ 𝐴} = 𝐴
4	3	ineq1i 4169	. 2 ⊢ ({𝑥 ∣ 𝑥 ∈ 𝐴} ∩ {𝑥 ∣ 𝜑}) = (𝐴 ∩ {𝑥 ∣ 𝜑})
5	1, 2, 4	3eqtr2i 2758	1 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = (𝐴 ∩ {𝑥 ∣ 𝜑})

Syntax hints:   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {cab 2707  {crab 3396   ∩ cin 3904

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3397  df-v 3440  df-in 3912

This theorem is referenced by:  dfrab2  4273  notrab  4275  dfrab3ss  4276  dfif3  4493  dffr3  6054  dfse2  6055  rabfi  9172  dfsup2  9353  ressmplbas2  21950  clsocv  25166  hasheuni  34054  bj-inrab3  36905  bj-reabeq  37003  hashnzfz  44296