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Mirrors > Home > MPE Home > Th. List > moabs | Structured version Visualization version 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 | pm5.4 377 | . 2 ⊢ ((∃𝑥𝜑 → (∃𝑥𝜑 → ∃!𝑥𝜑)) ↔ (∃𝑥𝜑 → ∃!𝑥𝜑)) | |
2 | df-mo 2475 | . . 3 ⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑)) | |
3 | 2 | imbi2i 326 | . 2 ⊢ ((∃𝑥𝜑 → ∃*𝑥𝜑) ↔ (∃𝑥𝜑 → (∃𝑥𝜑 → ∃!𝑥𝜑))) |
4 | 1, 3, 2 | 3bitr4ri 293 | 1 ⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃*𝑥𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∃wex 1704 ∃!weu 2470 ∃*wmo 2471 |
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 197 df-mo 2475 |
This theorem is referenced by: mo3 2507 dffun7 5915 bj-mo3OLD 32832 wl-mo3t 33358 |
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