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Theorem rmo5 3192
 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 df-mo 2503 . 2 (∃*𝑥(𝑥𝐴𝜑) ↔ (∃𝑥(𝑥𝐴𝜑) → ∃!𝑥(𝑥𝐴𝜑)))
2 df-rmo 2949 . 2 (∃*𝑥𝐴 𝜑 ↔ ∃*𝑥(𝑥𝐴𝜑))
3 df-rex 2947 . . 3 (∃𝑥𝐴 𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
4 df-reu 2948 . . 3 (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥(𝑥𝐴𝜑))
53, 4imbi12i 339 . 2 ((∃𝑥𝐴 𝜑 → ∃!𝑥𝐴 𝜑) ↔ (∃𝑥(𝑥𝐴𝜑) → ∃!𝑥(𝑥𝐴𝜑)))
61, 2, 53bitr4i 292 1 (∃*𝑥𝐴 𝜑 ↔ (∃𝑥𝐴 𝜑 → ∃!𝑥𝐴 𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383  ∃wex 1744   ∈ wcel 2030  ∃!weu 2498  ∃*wmo 2499  ∃wrex 2942  ∃!wreu 2943  ∃*wrmo 2944 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8 This theorem depends on definitions:  df-bi 197  df-mo 2503  df-rex 2947  df-reu 2948  df-rmo 2949 This theorem is referenced by:  nrexrmo  3193  cbvrmo  3200  ddemeas  30427  2reurex  41502  iccpartdisj  41698
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