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

Theorem rabbidva 2603
Description: Equivalent wff's yield equal restricted class abstractions (deduction rule). (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 2442 . 2 (𝜑 → ∀𝑥𝐴 (𝜓𝜒))
3 rabbi 2540 . 2 (∀𝑥𝐴 (𝜓𝜒) ↔ {𝑥𝐴𝜓} = {𝑥𝐴𝜒})
42, 3sylib 120 1 (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐴𝜒})
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103   = wceq 1287  wcel 1436  wral 2355  {crab 2359
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-11 1440  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-ral 2360  df-rab 2364
This theorem is referenced by:  rabbidv  2604  rabeqbidva  2611  rabbi2dva  3197  rabxfrd  4264  onsucmin  4296  seinxp  4476  fniniseg2  5378  fnniniseg2  5379  f1oresrab  5420  dfinfre  8345  minmax  10547  gcdass  10871  lcmass  10934
  Copyright terms: Public domain W3C validator