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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 3504 | . 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 1314 {csn 3493 |
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-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-11 1467 ax-4 1470 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 |
This theorem depends on definitions: df-bi 116 df-tru 1317 df-nf 1420 df-sb 1719 df-clab 2102 df-cleq 2108 df-sn 3499 |
This theorem is referenced by: dmsnsnsng 4974 cnvsng 4982 ressn 5037 f1osng 5364 fsng 5547 fnressn 5560 fvsng 5570 2nd1st 6032 dfmpo 6074 cnvf1olem 6075 tpostpos 6115 tfrlemi1 6183 tfr1onlemaccex 6199 tfrcllemaccex 6212 elixpsn 6583 ixpsnf1o 6584 en1bg 6648 mapsnen 6659 xpassen 6677 fztp 9751 fzsuc2 9752 fseq1p1m1 9767 fseq1m1p1 9768 zfz1isolemsplit 10474 zfz1isolem1 10476 fsumm1 11077 divalgmod 11472 ennnfonelemg 11761 ennnfonelemp1 11764 ennnfonelem1 11765 ennnfonelemnn0 11780 setsvalg 11832 strsetsid 11835 txdis 12288 |
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