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Theorem abeqin 37646
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 4204 . 2 (𝐵𝐶) = ({𝑥𝜑} ∩ 𝐶)
3 abeqin.1 . 2 𝐴 = (𝐵𝐶)
4 dfrab2 4306 . 2 {𝑥𝐶𝜑} = ({𝑥𝜑} ∩ 𝐶)
52, 3, 43eqtr4i 2765 1 𝐴 = {𝑥𝐶𝜑}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1534  {cab 2704  {crab 3427  cin 3943
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2698
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-rab 3428  df-v 3471  df-in 3951
This theorem is referenced by:  abeqinbi  37647  dfcnvrefrels3  37925  dffunsALTV  38079  dfdisjs  38104
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