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Theorem abeqinbi 38235
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 38234 . 2 𝐴 = {𝑥𝐶𝜑}
4 abeqinbi.3 . 2 (𝑥𝐶 → (𝜑𝜓))
53, 4rabimbieq 38233 1 𝐴 = {𝑥𝐶𝜓}
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1537  wcel 2106  {cab 2712  {crab 3433  cin 3962
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-in 3970
This theorem is referenced by:  dfrefrels2  38495  dfcnvrefrels2  38510  dfsymrels2  38527  dftrrels2  38557
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