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Theorem rabimbieq 35045
 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 3418 . 2 {𝑥𝐴𝜑} = {𝑥𝐴𝜓}
41, 3eqtri 2819 1 𝐵 = {𝑥𝐴𝜓}
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 207   = wceq 1522   ∈ wcel 2081  {crab 3109 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-9 2091  ax-ext 2769 This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1762  df-sb 2043  df-clab 2776  df-cleq 2788  df-rab 3114 This theorem is referenced by:  abeqinbi  35047  dfsymrels4  35314  dfsymrels5  35315  dffunsALTV2  35448  dffunsALTV3  35449  dffunsALTV4  35450  dfdisjs2  35473  dfdisjs5  35476
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