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Theorem rabimbieq 35394
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 3470 . 2 {𝑥𝐴𝜑} = {𝑥𝐴𝜓}
41, 3eqtri 2841 1 𝐵 = {𝑥𝐴𝜓}
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207   = wceq 1528  wcel 2105  {crab 3139
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-9 2115  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1772  df-sb 2061  df-clab 2797  df-cleq 2811  df-rab 3144
This theorem is referenced by:  abeqinbi  35396  dfsymrels4  35663  dfsymrels5  35664  dffunsALTV2  35797  dffunsALTV3  35798  dffunsALTV4  35799  dfdisjs2  35822  dfdisjs5  35825
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