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Theorem moabs 1965
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 pm5.4 242 . 2 ((∃𝑥𝜑 → (∃𝑥𝜑 → ∃!𝑥𝜑)) ↔ (∃𝑥𝜑 → ∃!𝑥𝜑))
2 df-mo 1920 . . 3 (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑))
32imbi2i 219 . 2 ((∃𝑥𝜑 → ∃*𝑥𝜑) ↔ (∃𝑥𝜑 → (∃𝑥𝜑 → ∃!𝑥𝜑)))
41, 3, 23bitr4ri 206 1 (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃*𝑥𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 102  wex 1397  ∃!weu 1916  ∃*wmo 1917
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105
This theorem depends on definitions:  df-bi 114  df-mo 1920
This theorem is referenced by:  mo2icl  2743  dffun7  4956
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