Theorem exmoeu 1415

Description: Existence in terms of "at most one" and uniqueness.

Assertion

Ref	Expression
exmoeu	⊢ (∃xφ ↔ (∃*xφ → ∃!xφ))

Proof of Theorem exmoeu

Step	Hyp	Ref	Expression
1		df-mo 1385	. . . 4 ⊢ (∃*xφ ↔ (∃xφ → ∃!xφ))
2	1	biimp 151	. . 3 ⊢ (∃*xφ → (∃xφ → ∃!xφ))
3	2	com12 11	. 2 ⊢ (∃xφ → (∃*xφ → ∃!xφ))
4	1	biimpr 152	. . . 4 ⊢ ((∃xφ → ∃!xφ) → ∃*xφ)
5		euex 1396	. . . 4 ⊢ (∃!xφ → ∃xφ)
6	4, 5	imim12i 18	. . 3 ⊢ ((∃*xφ → ∃!xφ) → ((∃xφ → ∃!xφ) → ∃xφ))
7		peirce 82	. . 3 ⊢ (((∃xφ → ∃!xφ) → ∃xφ) → ∃xφ)
8	6, 7	syl 10	. 2 ⊢ ((∃*xφ → ∃!xφ) → ∃xφ)
9	3, 8	impbi 157	1 ⊢ (∃xφ ↔ (∃*xφ → ∃!xφ))

Syntax hints:   → wi 3   ↔ wb 146  ∃wex 982  ∃!weu 1382  ∃*wmo 1383

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385

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