Theorem moabs 2248

Description: Absorption of existence condition by "at most one." (Contributed by NM, 4-Nov-2002.)

Assertion

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

Proof of Theorem moabs

Step	Hyp	Ref	Expression
1		pm5.4 351	. 2 ⊢ ((∃xφ → (∃xφ → ∃!xφ)) ↔ (∃xφ → ∃!xφ))
2		df-mo 2209	. . 3 ⊢ (∃*xφ ↔ (∃xφ → ∃!xφ))
3	2	imbi2i 303	. 2 ⊢ ((∃xφ → ∃*xφ) ↔ (∃xφ → (∃xφ → ∃!xφ)))
4	1, 3, 2	3bitr4ri 269	1 ⊢ (∃*xφ ↔ (∃xφ → ∃*xφ))

Syntax hints:   → wi 4   ↔ wb 176  ∃wex 1541  ∃!weu 2204  ∃*wmo 2205

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 177  df-mo 2209

This theorem is referenced by:  dffun7  5134