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Mirrors > Home > ILE Home > Th. List > moabs | GIF version |
Description: Absorption of existence condition by "at most one." (Contributed by NM, 4-Nov-2002.) |
Ref | Expression |
---|---|
moabs | ⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃*𝑥𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm5.4 248 | . 2 ⊢ ((∃𝑥𝜑 → (∃𝑥𝜑 → ∃!𝑥𝜑)) ↔ (∃𝑥𝜑 → ∃!𝑥𝜑)) | |
2 | df-mo 2010 | . . 3 ⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑)) | |
3 | 2 | imbi2i 225 | . 2 ⊢ ((∃𝑥𝜑 → ∃*𝑥𝜑) ↔ (∃𝑥𝜑 → (∃𝑥𝜑 → ∃!𝑥𝜑))) |
4 | 1, 3, 2 | 3bitr4ri 212 | 1 ⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃*𝑥𝜑)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∃wex 1472 ∃!weu 2006 ∃*wmo 2007 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 |
This theorem depends on definitions: df-bi 116 df-mo 2010 |
This theorem is referenced by: mo2icl 2891 dffun7 5197 |
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