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

Theorem rabeq2i 2548
 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 = {x Bφ}
Assertion
Ref Expression
rabeq2i (x A ↔ (x B φ))

Proof of Theorem rabeq2i
StepHypRef Expression
1 rabeqi.1 . . 3 A = {x Bφ}
21eleq2i 2101 . 2 (x Ax {x Bφ})
3 rabid 2479 . 2 (x {x Bφ} ↔ (x B φ))
42, 3bitri 173 1 (x A ↔ (x B φ))
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   ↔ wb 98   = wceq 1242   ∈ wcel 1390  {crab 2304 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-rab 2309 This theorem is referenced by:  tfis  4249  fvmptssdm  5198
 Copyright terms: Public domain W3C validator