Theorem dfrab2 4199

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 4198	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = (𝐴 ∩ {𝑥 ∣ 𝜑})
2		incom 4091	. 2 ⊢ (𝐴 ∩ {𝑥 ∣ 𝜑}) = ({𝑥 ∣ 𝜑} ∩ 𝐴)
3	1, 2	eqtri 2761	1 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = ({𝑥 ∣ 𝜑} ∩ 𝐴)

Syntax hints:   = wceq 1542  {cab 2716  {crab 3057   ∩ cin 3842

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-ext 2710

This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1545  df-ex 1787  df-sb 2075  df-clab 2717  df-cleq 2730  df-clel 2811  df-rab 3062  df-v 3400  df-in 3850

This theorem is referenced by:  dfpred3  6139  lubdm  17707  glbdm  17720  psrbagsn  20877  ismbl  24280  eulerpartgbij  31911  orvcval4  31999  lrrecse  33744  lrrecpred  33746  fvline2  34093  abeqin  36037  rabdif  39797  nznngen  41494