MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rmo5 Structured version   Visualization version   GIF version

Theorem rmo5 3399
Description: Restricted "at most one" in term of uniqueness. (Contributed by NM, 16-Jun-2017.)
Assertion
Ref Expression
rmo5 (∃*𝑥𝐴 𝜑 ↔ (∃𝑥𝐴 𝜑 → ∃!𝑥𝐴 𝜑))

Proof of Theorem rmo5
StepHypRef Expression
1 moeu 2582 . 2 (∃*𝑥(𝑥𝐴𝜑) ↔ (∃𝑥(𝑥𝐴𝜑) → ∃!𝑥(𝑥𝐴𝜑)))
2 df-rmo 3379 . 2 (∃*𝑥𝐴 𝜑 ↔ ∃*𝑥(𝑥𝐴𝜑))
3 df-rex 3070 . . 3 (∃𝑥𝐴 𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
4 df-reu 3380 . . 3 (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥(𝑥𝐴𝜑))
53, 4imbi12i 350 . 2 ((∃𝑥𝐴 𝜑 → ∃!𝑥𝐴 𝜑) ↔ (∃𝑥(𝑥𝐴𝜑) → ∃!𝑥(𝑥𝐴𝜑)))
61, 2, 53bitr4i 303 1 (∃*𝑥𝐴 𝜑 ↔ (∃𝑥𝐴 𝜑 → ∃!𝑥𝐴 𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wex 1778  wcel 2107  ∃*wmo 2537  ∃!weu 2567  wrex 3069  ∃!wreu 3377  ∃*wrmo 3378
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1779  df-mo 2539  df-eu 2568  df-rex 3070  df-rmo 3379  df-reu 3380
This theorem is referenced by:  nrexrmo  3400  cbvrmo  3428  2reurex  3765  rmo0  4361  rmosn  4718  ddemeas  34238  iccpartdisj  47429
  Copyright terms: Public domain W3C validator