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

Theorem rabbidva 2677
 Description: Equivalent wff's yield equal restricted class abstractions (deduction form). (Contributed by NM, 28-Nov-2003.)
Hypothesis
Ref Expression
rabbidva.1 ((𝜑𝑥𝐴) → (𝜓𝜒))
Assertion
Ref Expression
rabbidva (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐴𝜒})
Distinct variable group:   𝜑,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑥)   𝐴(𝑥)

Proof of Theorem rabbidva
StepHypRef Expression
1 rabbidva.1 . . 3 ((𝜑𝑥𝐴) → (𝜓𝜒))
21ralrimiva 2508 . 2 (𝜑 → ∀𝑥𝐴 (𝜓𝜒))
3 rabbi 2611 . 2 (∀𝑥𝐴 (𝜓𝜒) ↔ {𝑥𝐴𝜓} = {𝑥𝐴𝜒})
42, 3sylib 121 1 (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐴𝜒})
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1332   ∈ wcel 1481  ∀wral 2417  {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-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-11 1485  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122 This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-ral 2422  df-rab 2426 This theorem is referenced by:  rabbidv  2678  rabeqbidva  2685  rabbi2dva  3289  rabxfrd  4398  onsucmin  4431  seinxp  4618  fniniseg2  5550  fnniniseg2  5551  f1oresrab  5593  dfinfre  8739  minmax  11034  xrminmax  11067  iooinsup  11079  gcdass  11740  lcmass  11803  bdbl  12712  xmetxpbl  12717
 Copyright terms: Public domain W3C validator