ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  rabeq2i GIF version

Theorem rabeq2i 2638
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 2166 . 2 (𝑥𝐴𝑥 ∈ {𝑥𝐵𝜑})
3 rabid 2564 . 2 (𝑥 ∈ {𝑥𝐵𝜑} ↔ (𝑥𝐵𝜑))
42, 3bitri 183 1 (𝑥𝐴 ↔ (𝑥𝐵𝜑))
Colors of variables: wff set class
Syntax hints:  wa 103  wb 104   = wceq 1299  wcel 1448  {crab 2379
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1391  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-ext 2082
This theorem depends on definitions:  df-bi 116  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-rab 2384
This theorem is referenced by:  tfis  4435  fvmptssdm  5437
  Copyright terms: Public domain W3C validator