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Theorem abeqinbi 35047
 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 35046 . 2 𝐴 = {𝑥𝐶𝜑}
4 abeqinbi.3 . 2 (𝑥𝐶 → (𝜑𝜓))
53, 4rabimbieq 35045 1 𝐴 = {𝑥𝐶𝜓}
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 207   = wceq 1522   ∈ wcel 2081  {cab 2775  {crab 3109   ∩ cin 3858 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-ext 2769 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-rab 3114  df-v 3439  df-in 3866 This theorem is referenced by:  dfrefrels2  35284  dfcnvrefrels2  35297  dfsymrels2  35312  dftrrels2  35342
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