![]() |
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 3602 | . 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 1353 {csn 3591 |
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-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-11 1506 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-sn 3597 |
This theorem is referenced by: dmsnsnsng 5101 cnvsng 5109 ressn 5164 f1osng 5497 fsng 5684 fnressn 5697 fvsng 5707 2nd1st 6174 dfmpo 6217 cnvf1olem 6218 tpostpos 6258 tfrlemi1 6326 tfr1onlemaccex 6342 tfrcllemaccex 6355 elixpsn 6728 ixpsnf1o 6729 en1bg 6793 mapsnen 6804 xpassen 6823 fztp 10051 fzsuc2 10052 fseq1p1m1 10067 fseq1m1p1 10068 zfz1isolemsplit 10789 zfz1isolem1 10791 fsumm1 11395 fprodm1 11577 divalgmod 11902 ennnfonelemg 12374 ennnfonelemp1 12377 ennnfonelem1 12378 ennnfonelemnn0 12393 setsvalg 12462 strsetsid 12465 mulgval 12862 isunitd 13087 txdis 13410 |
Copyright terms: Public domain | W3C validator |