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Theorem abeqinbi 35989
 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 35988 . 2 𝐴 = {𝑥𝐶𝜑}
4 abeqinbi.3 . 2 (𝑥𝐶 → (𝜑𝜓))
53, 4rabimbieq 35987 1 𝐴 = {𝑥𝐶𝜓}
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   = wceq 1538   ∈ wcel 2111  {cab 2735  {crab 3074   ∩ cin 3859 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 2113  ax-9 2121  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-rab 3079  df-v 3411  df-in 3867 This theorem is referenced by:  dfrefrels2  36227  dfcnvrefrels2  36240  dfsymrels2  36255  dftrrels2  36285
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