MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  reubida Structured version   Visualization version   GIF version

Theorem reubida 3119
Description: Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by Mario Carneiro, 19-Nov-2016.)
Hypotheses
Ref Expression
reubida.1 𝑥𝜑
reubida.2 ((𝜑𝑥𝐴) → (𝜓𝜒))
Assertion
Ref Expression
reubida (𝜑 → (∃!𝑥𝐴 𝜓 ↔ ∃!𝑥𝐴 𝜒))

Proof of Theorem reubida
StepHypRef Expression
1 reubida.1 . . 3 𝑥𝜑
2 reubida.2 . . . 4 ((𝜑𝑥𝐴) → (𝜓𝜒))
32pm5.32da 672 . . 3 (𝜑 → ((𝑥𝐴𝜓) ↔ (𝑥𝐴𝜒)))
41, 3eubid 2486 . 2 (𝜑 → (∃!𝑥(𝑥𝐴𝜓) ↔ ∃!𝑥(𝑥𝐴𝜒)))
5 df-reu 2916 . 2 (∃!𝑥𝐴 𝜓 ↔ ∃!𝑥(𝑥𝐴𝜓))
6 df-reu 2916 . 2 (∃!𝑥𝐴 𝜒 ↔ ∃!𝑥(𝑥𝐴𝜒))
74, 5, 63bitr4g 303 1 (𝜑 → (∃!𝑥𝐴 𝜓 ↔ ∃!𝑥𝐴 𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  wnf 1706  wcel 1988  ∃!weu 2468  ∃!wreu 2911
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-12 2045
This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1703  df-nf 1708  df-eu 2472  df-reu 2916
This theorem is referenced by:  reubidva  3120  poimirlem25  33405  reuan  40943
  Copyright terms: Public domain W3C validator