Theorem exmoeudc 2089

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 2030	. . . 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 2056	. . . 4 ⊢ (∃!𝑥𝜑 → ∃𝑥𝜑)
6	4, 5	imim12i 59	. . 3 ⊢ ((∃*𝑥𝜑 → ∃!𝑥𝜑) → ((∃𝑥𝜑 → ∃!𝑥𝜑) → ∃𝑥𝜑))
7		peircedc 914	. . 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 834  ∃wex 1492  ∃!weu 2026  ∃*wmo 2027

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 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535

This theorem depends on definitions:  df-bi 117  df-dc 835  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030

This theorem is referenced by: (None)