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

Theorem rabbidva 2727
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 2550 . 2 (𝜑 → ∀𝑥𝐴 (𝜓𝜒))
3 rabbi 2655 . 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 1353  wcel 2148  wral 2455  {crab 2459
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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-ral 2460  df-rab 2464
This theorem is referenced by:  rabbidv  2728  rabeqbidva  2735  rabbi2dva  3345  rabxfrd  4471  onsucmin  4508  seinxp  4699  fniniseg2  5640  fnniniseg2  5641  f1oresrab  5683  dfinfre  8915  minmax  11240  xrminmax  11275  iooinsup  11287  gcdass  12018  lcmass  12087  pcneg  12326  bdbl  14042  xmetxpbl  14047
  Copyright terms: Public domain W3C validator