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

Theorem reubiia 2494
 Description: Formula-building rule for restricted existential quantifier (inference rule). (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 427 . . 3 ((𝑥𝐴𝜑) ↔ (𝑥𝐴𝜓))
32eubii 1909 . 2 (∃!𝑥(𝑥𝐴𝜑) ↔ ∃!𝑥(𝑥𝐴𝜓))
4 df-reu 2313 . 2 (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥(𝑥𝐴𝜑))
5 df-reu 2313 . 2 (∃!𝑥𝐴 𝜓 ↔ ∃!𝑥(𝑥𝐴𝜓))
63, 4, 53bitr4i 201 1 (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥𝐴 𝜓)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   ∈ wcel 1393  ∃!weu 1900  ∃!wreu 2308 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-17 1419  ax-ial 1427 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-eu 1903  df-reu 2313 This theorem is referenced by:  reubii  2495
 Copyright terms: Public domain W3C validator