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Theorem rabimbieq 38252
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 3440 . 2 {𝑥𝐴𝜑} = {𝑥𝐴𝜓}
41, 3eqtri 2765 1 𝐵 = {𝑥𝐴𝜓}
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1540  wcel 2108  {crab 3436
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-rab 3437
This theorem is referenced by:  abeqinbi  38254  dfsymrels4  38548  dfsymrels5  38549  dffunsALTV2  38685  dffunsALTV3  38686  dffunsALTV4  38687  dfdisjs2  38710  dfdisjs5  38713
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