Theorem moabs 2075

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		pm5.4 249	. 2 ⊢ ((∃𝑥𝜑 → (∃𝑥𝜑 → ∃!𝑥𝜑)) ↔ (∃𝑥𝜑 → ∃!𝑥𝜑))
2		df-mo 2030	. . 3 ⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑))
3	2	imbi2i 226	. 2 ⊢ ((∃𝑥𝜑 → ∃*𝑥𝜑) ↔ (∃𝑥𝜑 → (∃𝑥𝜑 → ∃!𝑥𝜑)))
4	1, 3, 2	3bitr4ri 213	1 ⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃*𝑥𝜑))

Syntax hints:   → wi 4   ↔ wb 105  ∃wex 1492  ∃!weu 2026  ∃*wmo 2027

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This theorem depends on definitions:  df-bi 117  df-mo 2030

This theorem is referenced by:  mo2icl  2918  dffun7  5245