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Theorem rmo5 2827
 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 2209 . 2 (∃*x(x A φ) ↔ (x(x A φ) → ∃!x(x A φ)))
2 df-rmo 2622 . 2 (∃*x A φ∃*x(x A φ))
3 df-rex 2620 . . 3 (x A φx(x A φ))
4 df-reu 2621 . . 3 (∃!x A φ∃!x(x A φ))
53, 4imbi12i 316 . 2 ((x A φ∃!x A φ) ↔ (x(x A φ) → ∃!x(x A φ)))
61, 2, 53bitr4i 268 1 (∃*x A φ ↔ (x A φ∃!x A φ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∃wex 1541   ∈ wcel 1710  ∃!weu 2204  ∃*wmo 2205  ∃wrex 2615  ∃!wreu 2616  ∃*wrmo 2617 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8 This theorem depends on definitions:  df-bi 177  df-mo 2209  df-rex 2620  df-reu 2621  df-rmo 2622 This theorem is referenced by:  nrexrmo  2828  cbvrmo  2834
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