Theorem dfrab2 4281

Description: Alternate definition of restricted class abstraction. (Contributed by NM, 20-Sep-2003.) (Proof shortened by BJ, 22-Apr-2019.)

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		dfrab3 4280	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = (𝐴 ∩ {𝑥 ∣ 𝜑})
2		incom 4170	. 2 ⊢ (𝐴 ∩ {𝑥 ∣ 𝜑}) = ({𝑥 ∣ 𝜑} ∩ 𝐴)
3	1, 2	eqtri 2792	1 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = ({𝑥 ∣ 𝜑} ∩ 𝐴)

Syntax hints:   = wceq 1567  {cab 2747  {crab 3423   ∩ cin 3912

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1570  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-rab 3424  df-v 3465  df-in 3920

This theorem is referenced by:  rabdif  4282  dfpred3  6310  predres  6337  lubdm  18401  glbdm  18414  psrbagsn  22179  ismbl  25650  lrrecse  28097  lrrecpred  28099  eulerpartgbij  34703  orvcval4  34792  fvline2  36533  abeqin  38788  nznngen  44911