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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 1528 | . 2 ⊢ Ⅎ𝑥𝜓 | |
2 | mo4.1 | . 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
3 | 1, 2 | mo4f 2086 | 1 ⊢ (∃*𝑥𝜑 ↔ ∀𝑥∀𝑦((𝜑 ∧ 𝜓) → 𝑥 = 𝑦)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∀wal 1351 ∃*wmo 2027 |
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 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 |
This theorem depends on definitions: df-bi 117 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 |
This theorem is referenced by: eu4 2088 rmo4 2932 dffun5r 5230 dffun6f 5231 fun11 5285 brprcneu 5510 dff13 5772 mpofun 5980 caovimo 6071 th3qlem1 6640 exmidmotap 7263 addnq0mo 7449 mulnq0mo 7450 addsrmo 7745 mulsrmo 7746 summodc 11394 prodmodc 11589 limcimo 14295 |
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