Theorem exmoeudc 2062

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 2003	. . . 4 ⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑))
2	1	biimpi 119	. . 3 ⊢ (∃*𝑥𝜑 → (∃𝑥𝜑 → ∃!𝑥𝜑))
3	2	com12 30	. 2 ⊢ (∃𝑥𝜑 → (∃*𝑥𝜑 → ∃!𝑥𝜑))
4	1	biimpri 132	. . . 4 ⊢ ((∃𝑥𝜑 → ∃!𝑥𝜑) → ∃*𝑥𝜑)
5		euex 2029	. . . 4 ⊢ (∃!𝑥𝜑 → ∃𝑥𝜑)
6	4, 5	imim12i 59	. . 3 ⊢ ((∃*𝑥𝜑 → ∃!𝑥𝜑) → ((∃𝑥𝜑 → ∃!𝑥𝜑) → ∃𝑥𝜑))
7		peircedc 899	. . 3 ⊢ (DECID ∃𝑥𝜑 → (((∃𝑥𝜑 → ∃!𝑥𝜑) → ∃𝑥𝜑) → ∃𝑥𝜑))
8	6, 7	syl5 32	. 2 ⊢ (DECID ∃𝑥𝜑 → ((∃*𝑥𝜑 → ∃!𝑥𝜑) → ∃𝑥𝜑))
9	3, 8	impbid2 142	1 ⊢ (DECID ∃𝑥𝜑 → (∃𝑥𝜑 ↔ (∃*𝑥𝜑 → ∃!𝑥𝜑)))

Syntax hints:   → wi 4   ↔ wb 104  DECID wdc 819  ∃wex 1468  ∃!weu 1999  ∃*wmo 2000

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515

This theorem depends on definitions:  df-bi 116  df-dc 820  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003

This theorem is referenced by: (None)