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Theorem rabeq2i 2686
Description: Inference from equality of a class variable and a restricted class abstraction. (Contributed by NM, 16-Feb-2004.)
Hypothesis
Ref Expression
rabeq2i.1 𝐴 = {𝑥𝐵𝜑}
Assertion
Ref Expression
rabeq2i (𝑥𝐴 ↔ (𝑥𝐵𝜑))

Proof of Theorem rabeq2i
StepHypRef Expression
1 rabeq2i.1 . . 3 𝐴 = {𝑥𝐵𝜑}
21eleq2i 2207 . 2 (𝑥𝐴𝑥 ∈ {𝑥𝐵𝜑})
3 rabid 2609 . 2 (𝑥 ∈ {𝑥𝐵𝜑} ↔ (𝑥𝐵𝜑))
42, 3bitri 183 1 (𝑥𝐴 ↔ (𝑥𝐵𝜑))
Colors of variables: wff set class
Syntax hints:  wa 103  wb 104   = wceq 1332  wcel 1481  {crab 2421
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1424  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-rab 2426
This theorem is referenced by:  tfis  4504  fvmptssdm  5512  suplocsrlempr  7638  suplocsrlem  7639
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