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

Theorem reubidva 2613
Description: Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 13-Nov-2004.)
Hypothesis
Ref Expression
reubidva.1 ((𝜑𝑥𝐴) → (𝜓𝜒))
Assertion
Ref Expression
reubidva (𝜑 → (∃!𝑥𝐴 𝜓 ↔ ∃!𝑥𝐴 𝜒))
Distinct variable group:   𝜑,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑥)   𝐴(𝑥)

Proof of Theorem reubidva
StepHypRef Expression
1 nfv 1508 . 2 𝑥𝜑
2 reubidva.1 . 2 ((𝜑𝑥𝐴) → (𝜓𝜒))
31, 2reubida 2612 1 (𝜑 → (∃!𝑥𝐴 𝜓 ↔ ∃!𝑥𝐴 𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wcel 1480  ∃!wreu 2418
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 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-17 1506  ax-ial 1514
This theorem depends on definitions:  df-bi 116  df-nf 1437  df-eu 2002  df-reu 2423
This theorem is referenced by:  reubidv  2614  f1ofveu  5762  srpospr  7598  icoshftf1o  9781  divalgb  11629
  Copyright terms: Public domain W3C validator