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Theorem dfeumo 2613
Description: An elementary proof showing the reverse direction of dfmoeu 2612. Here the characterizing expression of existential uniqueness (eu6 2653) is derived from that of uniqueness (df-mo 2616). (Contributed by Wolf Lammen, 3-Oct-2023.)
Assertion
Ref Expression
dfeumo ((∃𝑥𝜑 ∧ ∃𝑦𝑥(𝜑𝑥 = 𝑦)) ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
Distinct variable groups:   𝑥,𝑦   𝜑,𝑦
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem dfeumo
StepHypRef Expression
1 ax6ev 1965 . . . . 5 𝑥 𝑥 = 𝑦
2 biimpr 222 . . . . . 6 ((𝜑𝑥 = 𝑦) → (𝑥 = 𝑦𝜑))
32aleximi 1825 . . . . 5 (∀𝑥(𝜑𝑥 = 𝑦) → (∃𝑥 𝑥 = 𝑦 → ∃𝑥𝜑))
41, 3mpi 20 . . . 4 (∀𝑥(𝜑𝑥 = 𝑦) → ∃𝑥𝜑)
54exlimiv 1924 . . 3 (∃𝑦𝑥(𝜑𝑥 = 𝑦) → ∃𝑥𝜑)
65pm4.71ri 563 . 2 (∃𝑦𝑥(𝜑𝑥 = 𝑦) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝑥(𝜑𝑥 = 𝑦)))
7 abai 824 . 2 ((∃𝑥𝜑 ∧ ∃𝑦𝑥(𝜑𝑥 = 𝑦)) ↔ (∃𝑥𝜑 ∧ (∃𝑥𝜑 → ∃𝑦𝑥(𝜑𝑥 = 𝑦))))
8 dfmoeu 2612 . . 3 ((∃𝑥𝜑 → ∃𝑦𝑥(𝜑𝑥 = 𝑦)) ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
98anbi2i 624 . 2 ((∃𝑥𝜑 ∧ (∃𝑥𝜑 → ∃𝑦𝑥(𝜑𝑥 = 𝑦))) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝑥(𝜑𝑥 = 𝑦)))
106, 7, 93bitrri 300 1 ((∃𝑥𝜑 ∧ ∃𝑦𝑥(𝜑𝑥 = 𝑦)) ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  wal 1528  wex 1773
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-10 2138  ax-12 2169
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1774  df-nf 1778
This theorem is referenced by: (None)
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