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Mirrors > Home > NFE 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 3744 | . 2 ⊢ (A = B → {A} = {B}) | |
3 | 1, 2 | syl 15 | 1 ⊢ (φ → {A} = {B}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1642 {csn 3737 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-an 360 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-sn 3741 |
This theorem is referenced by: elp6 4263 pw1eqadj 4332 elimapw13 4946 dmsnopg 5066 fnunsn 5190 fsng 5433 fnressn 5438 fvsng 5446 freceq12 6311 |
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