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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 2070 | . 2 ⊢ (∃!𝑥𝜓 → ∃*𝑥𝜓) | |
2 | moim 2102 | . 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 1362 ∃!weu 2038 ∃*wmo 2039 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 |
This theorem depends on definitions: df-bi 117 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 |
This theorem is referenced by: euim 2106 2eumo 2126 reuss2 3430 |
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