Theorem exmoeu 1039

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 1010	. . . 4 ⊢ (∃*xφ ↔ (∃xφ → ∃!xO/FONT>φ))
2	1	biimp 133	. . 3 ⊢ (∃*xφ → (∃xφ → ∃!xφ))
3	2	com12 13	. 2 ⊢ (∃xφ → (∃*xφ → ∃!xφ))
4	1	biimpr 134	. . . 4 ⊢ ((∃xφ → ∃!xφ) → ∃*xφ)
5		euex 1021	. . . 4 ⊢ (∃!xφ → ∃xφ)
6	4, 5	syl34 20	. . 3 ⊢ ((∃*xφ → ∃!xφ) → ((∃xφ → ∃!xφ) → ∃xφ))
7		peirce 76	. . 3 ⊢ (((∃xφ → ∃!xφ) → ∃xφ) → ∃xφ)
8	6, 7	syl 12	. 2 ⊢ ((∃*xφ → ∃!xφ) → ∃xφ)
9	3, 8	impbi 139	1 ⊢ (∃xφ ↔ (∃*xφ → ∃!xφ))

Syntax hints:   → wi 2   ↔ wb 127  ∃wex 678  ∃!weu 1007  ∃*wmo 1008

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010

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