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Theorem rabeq2i 2571
 Description: Inference rule from equality of a class variable and a restricted class abstraction. (Contributed by NM, 16-Feb-2004.)
Hypothesis
Ref Expression
rabeqi.1 𝐴 = {𝑥𝐵𝜑}
Assertion
Ref Expression
rabeq2i (𝑥𝐴 ↔ (𝑥𝐵𝜑))

Proof of Theorem rabeq2i
StepHypRef Expression
1 rabeqi.1 . . 3 𝐴 = {𝑥𝐵𝜑}
21eleq2i 2120 . 2 (𝑥𝐴𝑥 ∈ {𝑥𝐵𝜑})
3 rabid 2502 . 2 (𝑥 ∈ {𝑥𝐵𝜑} ↔ (𝑥𝐵𝜑))
42, 3bitri 177 1 (𝑥𝐴 ↔ (𝑥𝐵𝜑))
 Colors of variables: wff set class Syntax hints:   ∧ wa 101   ↔ wb 102   = wceq 1259   ∈ wcel 1409  {crab 2327 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-ext 2038 This theorem depends on definitions:  df-bi 114  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-rab 2332 This theorem is referenced by:  tfis  4333  fvmptssdm  5282
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