Theorem dfrab2 4270

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 4269	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = (𝐴 ∩ {𝑥 ∣ 𝜑})
2		incom 4159	. 2 ⊢ (𝐴 ∩ {𝑥 ∣ 𝜑}) = ({𝑥 ∣ 𝜑} ∩ 𝐴)
3	1, 2	eqtri 2784	1 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = ({𝑥 ∣ 𝜑} ∩ 𝐴)

Syntax hints:   = wceq 1559  {cab 2739  {crab 3413   ∩ cin 3901

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1562  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-rab 3414  df-v 3455  df-in 3909

This theorem is referenced by:  rabdif  4271  dfpred3  6294  predres  6321  lubdm  18372  glbdm  18385  psrbagsn  22104  ismbl  25576  lrrecse  28023  lrrecpred  28025  eulerpartgbij  34630  orvcval4  34719  fvline2  36457  abeqin  38714  nznngen  44853