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Theorem moabs 2500
Description: Absorption of existence condition by "at most one." (Contributed by NM, 4-Nov-2002.)
Assertion
Ref Expression
moabs (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃*𝑥𝜑))

Proof of Theorem moabs
StepHypRef Expression
1 pm5.4 377 . 2 ((∃𝑥𝜑 → (∃𝑥𝜑 → ∃!𝑥𝜑)) ↔ (∃𝑥𝜑 → ∃!𝑥𝜑))
2 df-mo 2474 . . 3 (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑))
32imbi2i 326 . 2 ((∃𝑥𝜑 → ∃*𝑥𝜑) ↔ (∃𝑥𝜑 → (∃𝑥𝜑 → ∃!𝑥𝜑)))
41, 3, 23bitr4ri 293 1 (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃*𝑥𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wex 1701  ∃!weu 2469  ∃*wmo 2470
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 2474
This theorem is referenced by:  mo3  2506  dffun7  5879  bj-mo3OLD  32504  wl-mo3t  33017
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