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Theorem abeqinbi 36526
Description: Intersection with class abstraction and equivalent wff's. (Contributed by Peter Mazsa, 21-Jul-2021.)
Hypotheses
Ref Expression
abeqinbi.1 𝐴 = (𝐵𝐶)
abeqinbi.2 𝐵 = {𝑥𝜑}
abeqinbi.3 (𝑥𝐶 → (𝜑𝜓))
Assertion
Ref Expression
abeqinbi 𝐴 = {𝑥𝐶𝜓}
Distinct variable group:   𝑥,𝐶
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)   𝐴(𝑥)   𝐵(𝑥)

Proof of Theorem abeqinbi
StepHypRef Expression
1 abeqinbi.1 . . 3 𝐴 = (𝐵𝐶)
2 abeqinbi.2 . . 3 𝐵 = {𝑥𝜑}
31, 2abeqin 36525 . 2 𝐴 = {𝑥𝐶𝜑}
4 abeqinbi.3 . 2 (𝑥𝐶 → (𝜑𝜓))
53, 4rabimbieq 36524 1 𝐴 = {𝑥𝐶𝜓}
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1540  wcel 2105  {cab 2713  {crab 3403  cin 3897
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-rab 3404  df-v 3443  df-in 3905
This theorem is referenced by:  dfrefrels2  36788  dfcnvrefrels2  36803  dfsymrels2  36820  dftrrels2  36850
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