Mathbox for Peter Mazsa < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  abeqin Structured version   Visualization version   GIF version

Theorem abeqin 35667
 Description: Intersection with class abstraction. (Contributed by Peter Mazsa, 21-Jul-2021.)
Hypotheses
Ref Expression
abeqin.1 𝐴 = (𝐵𝐶)
abeqin.2 𝐵 = {𝑥𝜑}
Assertion
Ref Expression
abeqin 𝐴 = {𝑥𝐶𝜑}
Distinct variable group:   𝑥,𝐶
Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥)   𝐵(𝑥)

Proof of Theorem abeqin
StepHypRef Expression
1 abeqin.2 . . 3 𝐵 = {𝑥𝜑}
21ineq1i 4138 . 2 (𝐵𝐶) = ({𝑥𝜑} ∩ 𝐶)
3 abeqin.1 . 2 𝐴 = (𝐵𝐶)
4 dfrab2 4234 . 2 {𝑥𝐶𝜑} = ({𝑥𝜑} ∩ 𝐶)
52, 3, 43eqtr4i 2834 1 𝐴 = {𝑥𝐶𝜑}
 Colors of variables: wff setvar class Syntax hints:   = wceq 1538  {cab 2779  {crab 3113   ∩ cin 3883 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-ext 2773 This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-rab 3118  df-v 3446  df-in 3891 This theorem is referenced by:  abeqinbi  35668  dfcnvrefrels3  35920  dffunsALTV  36069  dfdisjs  36094
 Copyright terms: Public domain W3C validator