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

Theorem reubiia 2613
 Description: Formula-building rule for restricted existential quantifier (inference form). (Contributed by NM, 14-Nov-2004.)
Hypothesis
Ref Expression
reubiia.1 (𝑥𝐴 → (𝜑𝜓))
Assertion
Ref Expression
reubiia (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥𝐴 𝜓)

Proof of Theorem reubiia
StepHypRef Expression
1 reubiia.1 . . . 4 (𝑥𝐴 → (𝜑𝜓))
21pm5.32i 449 . . 3 ((𝑥𝐴𝜑) ↔ (𝑥𝐴𝜓))
32eubii 2006 . 2 (∃!𝑥(𝑥𝐴𝜑) ↔ ∃!𝑥(𝑥𝐴𝜓))
4 df-reu 2421 . 2 (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥(𝑥𝐴𝜑))
5 df-reu 2421 . 2 (∃!𝑥𝐴 𝜓 ↔ ∃!𝑥(𝑥𝐴𝜓))
63, 4, 53bitr4i 211 1 (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥𝐴 𝜓)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∈ wcel 1480  ∃!weu 1997  ∃!wreu 2416 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-tru 1334  df-nf 1437  df-eu 2000  df-reu 2421 This theorem is referenced by:  reubii  2614
 Copyright terms: Public domain W3C validator