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Theorem dfmo 2596
Description: Rederive df-mo 2540 from the old definition moeu 2583. (Contributed by Wolf Lammen, 27-May-2019.) (Proof modification is discouraged.) Use df-mo 2540 instead. (New usage is discouraged.)
Assertion
Ref Expression
dfmo (∃*𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
Distinct variable groups:   𝑥,𝑦   𝜑,𝑦
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem dfmo
StepHypRef Expression
1 moeu 2583 . 2 (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑))
2 eu6 2574 . . 3 (∃!𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
32imbi2i 336 . 2 ((∃𝑥𝜑 → ∃!𝑥𝜑) ↔ (∃𝑥𝜑 → ∃𝑦𝑥(𝜑𝑥 = 𝑦)))
4 dfmoeu 2536 . 2 ((∃𝑥𝜑 → ∃𝑦𝑥(𝜑𝑥 = 𝑦)) ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
51, 3, 43bitri 297 1 (∃*𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wal 1537  wex 1782  ∃*wmo 2538  ∃!weu 2568
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-10 2137  ax-12 2171
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-ex 1783  df-nf 1787  df-mo 2540  df-eu 2569
This theorem is referenced by: (None)
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