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Theorem moabs 2075
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 pm5.4 249 . 2 ((∃𝑥𝜑 → (∃𝑥𝜑 → ∃!𝑥𝜑)) ↔ (∃𝑥𝜑 → ∃!𝑥𝜑))
2 df-mo 2030 . . 3 (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑))
32imbi2i 226 . 2 ((∃𝑥𝜑 → ∃*𝑥𝜑) ↔ (∃𝑥𝜑 → (∃𝑥𝜑 → ∃!𝑥𝜑)))
41, 3, 23bitr4ri 213 1 (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃*𝑥𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105  wex 1492  ∃!weu 2026  ∃*wmo 2027
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108
This theorem depends on definitions:  df-bi 117  df-mo 2030
This theorem is referenced by:  mo2icl  2916  dffun7  5243
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