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Theorem exmoeuOLD 2651
 Description: Obsolete proof of exmoeu 2621 as of 7-Oct-2022. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Wolf Lammen, 5-Dec-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
exmoeuOLD (∃𝑥𝜑 ↔ (∃*𝑥𝜑 → ∃!𝑥𝜑))

Proof of Theorem exmoeuOLD
StepHypRef Expression
1 moeu 2623 . . . 4 (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑))
21biimpi 208 . . 3 (∃*𝑥𝜑 → (∃𝑥𝜑 → ∃!𝑥𝜑))
32com12 32 . 2 (∃𝑥𝜑 → (∃*𝑥𝜑 → ∃!𝑥𝜑))
4 exmo 2602 . . . . 5 (∃𝑥𝜑 ∨ ∃*𝑥𝜑)
54ori 888 . . . 4 (¬ ∃𝑥𝜑 → ∃*𝑥𝜑)
65con1i 147 . . 3 (¬ ∃*𝑥𝜑 → ∃𝑥𝜑)
7 euex 2617 . . 3 (∃!𝑥𝜑 → ∃𝑥𝜑)
86, 7ja 175 . 2 ((∃*𝑥𝜑 → ∃!𝑥𝜑) → ∃𝑥𝜑)
93, 8impbii 201 1 (∃𝑥𝜑 ↔ (∃*𝑥𝜑 → ∃!𝑥𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198  ∃wex 1875  ∃*wmo 2589  ∃!weu 2608 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072 This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-ex 1876  df-mo 2591  df-eu 2609 This theorem is referenced by: (None)
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