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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 | pm5.4 238 | . 2 ⊢ ((∃𝑥𝜑 → (∃𝑥𝜑 → ∃!𝑥𝜑)) ↔ (∃𝑥𝜑 → ∃!𝑥𝜑)) | |
2 | df-mo 1904 | . . 3 ⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑)) | |
3 | 2 | imbi2i 215 | . 2 ⊢ ((∃𝑥𝜑 → ∃*𝑥𝜑) ↔ (∃𝑥𝜑 → (∃𝑥𝜑 → ∃!𝑥𝜑))) |
4 | 1, 3, 2 | 3bitr4ri 202 | 1 ⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃*𝑥𝜑)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 98 ∃wex 1381 ∃!weu 1900 ∃*wmo 1901 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 |
This theorem depends on definitions: df-bi 110 df-mo 1904 |
This theorem is referenced by: mo2icl 2720 dffun7 4928 |
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