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Mirrors > Home > ILE Home > Th. List > 2eumo | GIF version |
Description: Double quantification with existential uniqueness and "at most one." (Contributed by NM, 3-Dec-2001.) |
Ref | Expression |
---|---|
2eumo | ⊢ (∃!𝑥∃*𝑦𝜑 → ∃*𝑥∃!𝑦𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | euimmo 2066 | . 2 ⊢ (∀𝑥(∃!𝑦𝜑 → ∃*𝑦𝜑) → (∃!𝑥∃*𝑦𝜑 → ∃*𝑥∃!𝑦𝜑)) | |
2 | eumo 2031 | . 2 ⊢ (∃!𝑦𝜑 → ∃*𝑦𝜑) | |
3 | 1, 2 | mpg 1427 | 1 ⊢ (∃!𝑥∃*𝑦𝜑 → ∃*𝑥∃!𝑦𝜑) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∃!weu 1999 ∃*wmo 2000 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 |
This theorem depends on definitions: df-bi 116 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 |
This theorem is referenced by: (None) |
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