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Mirrors > Home > ILE Home > Th. List > mo4 | GIF version |
Description: "At most one" expressed using implicit substitution. (Contributed by NM, 26-Jul-1995.) |
Ref | Expression |
---|---|
mo4.1 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
mo4 | ⊢ (∃*𝑥𝜑 ↔ ∀𝑥∀𝑦((𝜑 ∧ 𝜓) → 𝑥 = 𝑦)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1473 | . 2 ⊢ Ⅎ𝑥𝜓 | |
2 | mo4.1 | . 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
3 | 1, 2 | mo4f 2015 | 1 ⊢ (∃*𝑥𝜑 ↔ ∀𝑥∀𝑦((𝜑 ∧ 𝜓) → 𝑥 = 𝑦)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∀wal 1294 ∃*wmo 1956 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 |
This theorem depends on definitions: df-bi 116 df-nf 1402 df-sb 1700 df-eu 1958 df-mo 1959 |
This theorem is referenced by: eu4 2017 rmo4 2822 dffun5r 5061 dffun6f 5062 fun11 5115 brprcneu 5333 dff13 5585 mpt2fun 5785 caovimo 5876 th3qlem1 6434 addnq0mo 7103 mulnq0mo 7104 addsrmo 7386 mulsrmo 7387 summodc 10941 |
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