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Mirrors > Home > ILE Home > Th. List > preq2 | GIF version |
Description: Equality theorem for unordered pairs. (Contributed by NM, 5-Aug-1993.) |
Ref | Expression |
---|---|
preq2 | ⊢ (A = B → {𝐶, A} = {𝐶, B}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | preq1 3438 | . 2 ⊢ (A = B → {A, 𝐶} = {B, 𝐶}) | |
2 | prcom 3437 | . 2 ⊢ {𝐶, A} = {A, 𝐶} | |
3 | prcom 3437 | . 2 ⊢ {𝐶, B} = {B, 𝐶} | |
4 | 1, 2, 3 | 3eqtr4g 2094 | 1 ⊢ (A = B → {𝐶, A} = {𝐶, B}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1242 {cpr 3368 |
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-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-v 2553 df-un 2916 df-sn 3373 df-pr 3374 |
This theorem is referenced by: preq12 3440 preq2i 3442 preq2d 3445 tpeq2 3448 preq12bg 3535 opeq2 3541 uniprg 3586 intprg 3639 prexgOLD 3937 prexg 3938 opth 3965 opeqsn 3980 relop 4429 funopg 4877 bj-prexg 9366 |
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