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Theorem rabimbieq 34359
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 3334 . 2 {𝑥𝐴𝜑} = {𝑥𝐴𝜓}
41, 3eqtri 2793 1 𝐵 = {𝑥𝐴𝜓}
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1631  wcel 2145  {crab 3065
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-rab 3070
This theorem is referenced by:  abeqinbi  34361  dfsymrels4  34635  dfsymrels5  34636
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