Theorem moabs 1041

Description: Absorption of existence condition by "at most one".

Assertion

Ref	Expression
moabs	⊢ (∃*xφ ↔ (∃xφ → ∃*xφ))

Proof of Theorem moabs

Step	Hyp	Ref	Expression
1		pm5.4 146	. 2 ⊢ ((∃xφ → (∃xφ → ∃!xφ)) ↔ (∃xφ → ∃!xφ))
2		df-mo 1010	. . 3 ⊢ (∃*xφ ↔ (∃xφ → ∃!xφ))
3	2	imbi2i 160	. 2 ⊢ ((∃xφ → ∃*xφ) ↔ (∃xφ → (∃xφ → ∃!xφ)))
4	1, 3, 2	3bitr4r 159	1 ⊢ (∃*xφ ↔ (∃xφ → ∃*xφ))

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

This theorem is referenced by:  dffun6 2687

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6

This theorem depends on definitions:  df-bi 128  df-mo 1010

metamath.org