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Mirrors > Home > ILE Home > Th. List > euimmo | GIF version |
Description: Uniqueness implies "at most one" through implication. (Contributed by NM, 22-Apr-1995.) |
Ref | Expression |
---|---|
euimmo | ⊢ (∀𝑥(𝜑 → 𝜓) → (∃!𝑥𝜓 → ∃*𝑥𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eumo 1980 | . 2 ⊢ (∃!𝑥𝜓 → ∃*𝑥𝜓) | |
2 | moim 2012 | . 2 ⊢ (∀𝑥(𝜑 → 𝜓) → (∃*𝑥𝜓 → ∃*𝑥𝜑)) | |
3 | 1, 2 | syl5 32 | 1 ⊢ (∀𝑥(𝜑 → 𝜓) → (∃!𝑥𝜓 → ∃*𝑥𝜑)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1287 ∃!weu 1948 ∃*wmo 1949 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 |
This theorem depends on definitions: df-bi 115 df-nf 1395 df-sb 1693 df-eu 1951 df-mo 1952 |
This theorem is referenced by: euim 2016 2eumo 2036 reuss2 3279 |
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