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Theorem rmo5 2499
 Description: Restricted "at most one" in term of uniqueness. (Contributed by NM, 16-Jun-2017.)
Assertion
Ref Expression
rmo5 (∃*x A φ ↔ (x A φ∃!x A φ))

Proof of Theorem rmo5
StepHypRef Expression
1 df-mo 1882 . 2 (∃*x(x A φ) ↔ (x(x A φ) → ∃!x(x A φ)))
2 df-rmo 2288 . 2 (∃*x A φ∃*x(x A φ))
3 df-rex 2286 . . 3 (x A φx(x A φ))
4 df-reu 2287 . . 3 (∃!x A φ∃!x(x A φ))
53, 4imbi12i 228 . 2 ((x A φ∃!x A φ) ↔ (x(x A φ) → ∃!x(x A φ)))
61, 2, 53bitr4i 201 1 (∃*x A φ ↔ (x A φ∃!x A φ))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98  ∃wex 1358   ∈ wcel 1370  ∃!weu 1878  ∃*wmo 1879  ∃wrex 2281  ∃!wreu 2282  ∃*wrmo 2283 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101 This theorem depends on definitions:  df-bi 110  df-mo 1882  df-rex 2286  df-reu 2287  df-rmo 2288 This theorem is referenced by:  nrexrmo  2500  cbvrmo  2506  bdrmo  7302
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