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Theorem rabeq2i 2746
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 2254 . 2 (𝑥𝐴𝑥 ∈ {𝑥𝐵𝜑})
3 rabid 2663 . 2 (𝑥 ∈ {𝑥𝐵𝜑} ↔ (𝑥𝐵𝜑))
42, 3bitri 184 1 (𝑥𝐴 ↔ (𝑥𝐵𝜑))
Colors of variables: wff set class
Syntax hints:  wa 104  wb 105   = wceq 1363  wcel 2158  {crab 2469
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1457  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-ext 2169
This theorem depends on definitions:  df-bi 117  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-rab 2474
This theorem is referenced by:  tfis  4594  fvmptssdm  5613  suplocsrlempr  7820  suplocsrlem  7821
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