Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > sneqd | GIF version |
Description: Equality deduction for singletons. (Contributed by NM, 22-Jan-2004.) |
Ref | Expression |
---|---|
sneqd.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
sneqd | ⊢ (𝜑 → {𝐴} = {𝐵}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sneqd.1 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | sneq 3386 | . 2 ⊢ (𝐴 = 𝐵 → {𝐴} = {𝐵}) | |
3 | 1, 2 | syl 14 | 1 ⊢ (𝜑 → {𝐴} = {𝐵}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1243 {csn 3375 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-11 1397 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-sn 3381 |
This theorem is referenced by: dmsnsnsng 4798 cnvsng 4806 ressn 4858 f1osng 5167 fsng 5336 fnressn 5349 fvsng 5359 2nd1st 5806 dfmpt2 5844 cnvf1olem 5845 tpostpos 5879 tfrlemi1 5946 en1bg 6280 xpassen 6304 fztp 8940 fzsuc2 8941 fseq1p1m1 8956 fseq1m1p1 8957 |
Copyright terms: Public domain | W3C validator |