Theorem exmoeudc 2143

Description: Existence in terms of "at most one" and uniqueness. (Contributed by Jim Kingdon, 3-Jul-2018.)

Assertion

Ref	Expression
exmoeudc	⊢ (DECID ∃𝑥𝜑 → (∃𝑥𝜑 ↔ (∃*𝑥𝜑 → ∃!𝑥𝜑)))

Proof of Theorem exmoeudc

Step	Hyp	Ref	Expression
1		df-mo 2083	. . . 4 ⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑))
2	1	biimpi 120	. . 3 ⊢ (∃*𝑥𝜑 → (∃𝑥𝜑 → ∃!𝑥𝜑))
3	2	com12 30	. 2 ⊢ (∃𝑥𝜑 → (∃*𝑥𝜑 → ∃!𝑥𝜑))
4	1	biimpri 133	. . . 4 ⊢ ((∃𝑥𝜑 → ∃!𝑥𝜑) → ∃*𝑥𝜑)
5		euex 2109	. . . 4 ⊢ (∃!𝑥𝜑 → ∃𝑥𝜑)
6	4, 5	imim12i 59	. . 3 ⊢ ((∃*𝑥𝜑 → ∃!𝑥𝜑) → ((∃𝑥𝜑 → ∃!𝑥𝜑) → ∃𝑥𝜑))
7		peircedc 921	. . 3 ⊢ (DECID ∃𝑥𝜑 → (((∃𝑥𝜑 → ∃!𝑥𝜑) → ∃𝑥𝜑) → ∃𝑥𝜑))
8	6, 7	syl5 32	. 2 ⊢ (DECID ∃𝑥𝜑 → ((∃*𝑥𝜑 → ∃!𝑥𝜑) → ∃𝑥𝜑))
9	3, 8	impbid2 143	1 ⊢ (DECID ∃𝑥𝜑 → (∃𝑥𝜑 ↔ (∃*𝑥𝜑 → ∃!𝑥𝜑)))

Syntax hints:   → wi 4   ↔ wb 105  DECID wdc 841  ∃wex 1540  ∃!weu 2079  ∃*wmo 2080

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583

This theorem depends on definitions:  df-bi 117  df-dc 842  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083

This theorem is referenced by: (None)