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Theorem rabimbieq 37855
Description: Restricted equivalent wff's correspond to restricted class abstractions which are equal with the same class. (Contributed by Peter Mazsa, 22-Jul-2021.)
Hypotheses
Ref Expression
rabimbieq.1 𝐵 = {𝑥𝐴𝜑}
rabimbieq.2 (𝑥𝐴 → (𝜑𝜓))
Assertion
Ref Expression
rabimbieq 𝐵 = {𝑥𝐴𝜓}

Proof of Theorem rabimbieq
StepHypRef Expression
1 rabimbieq.1 . 2 𝐵 = {𝑥𝐴𝜑}
2 rabimbieq.2 . . 3 (𝑥𝐴 → (𝜑𝜓))
32rabbiia 3422 . 2 {𝑥𝐴𝜑} = {𝑥𝐴𝜓}
41, 3eqtri 2753 1 𝐵 = {𝑥𝐴𝜓}
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1533  wcel 2098  {crab 3418
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-9 2108  ax-ext 2696
This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-rab 3419
This theorem is referenced by:  abeqinbi  37857  dfsymrels4  38151  dfsymrels5  38152  dffunsALTV2  38288  dffunsALTV3  38289  dffunsALTV4  38290  dfdisjs2  38313  dfdisjs5  38316
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