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

Theorem rabbidva 2740
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 2563 . 2 (𝜑 → ∀𝑥𝐴 (𝜓𝜒))
3 rabbi 2668 . 2 (∀𝑥𝐴 (𝜓𝜒) ↔ {𝑥𝐴𝜓} = {𝑥𝐴𝜒})
42, 3sylib 122 1 (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐴𝜒})
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1364  wcel 2160  wral 2468  {crab 2472
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-ral 2473  df-rab 2477
This theorem is referenced by:  rabbidv  2741  rabeqbidva  2748  rabbi2dva  3358  rabxfrd  4487  onsucmin  4524  seinxp  4715  fniniseg2  5659  fnniniseg2  5660  f1oresrab  5702  dfinfre  8943  minmax  11270  xrminmax  11305  iooinsup  11317  gcdass  12048  lcmass  12117  pcneg  12357  bdbl  14463  xmetxpbl  14468
  Copyright terms: Public domain W3C validator