Theorem dfrab3 4271

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 3400	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
2		inab 4261	. 2 ⊢ ({𝑥 ∣ 𝑥 ∈ 𝐴} ∩ {𝑥 ∣ 𝜑}) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
3		abid2 2873	. . 3 ⊢ {𝑥 ∣ 𝑥 ∈ 𝐴} = 𝐴
4	3	ineq1i 4168	. 2 ⊢ ({𝑥 ∣ 𝑥 ∈ 𝐴} ∩ {𝑥 ∣ 𝜑}) = (𝐴 ∩ {𝑥 ∣ 𝜑})
5	1, 2, 4	3eqtr2i 2765	1 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = (𝐴 ∩ {𝑥 ∣ 𝜑})

Syntax hints:   ∧ wa 395   = wceq 1541   ∈ wcel 2113  {cab 2714  {crab 3399   ∩ cin 3900

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-rab 3400  df-v 3442  df-in 3908

This theorem is referenced by:  dfrab2  4272  notrab  4274  dfrab3ss  4275  dfif3  4494  dffr3  6058  dfse2  6059  rabfi  9171  dfsup2  9347  ressmplbas2  21982  clsocv  25206  hasheuni  34242  bj-inrab3  37130  bj-reabeq  37228  hashnzfz  44561