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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 3570 | . 2 ⊢ (𝐴 = 𝐵 → {𝐴, 𝐶} = {𝐵, 𝐶}) | |
2 | prcom 3569 | . 2 ⊢ {𝐶, 𝐴} = {𝐴, 𝐶} | |
3 | prcom 3569 | . 2 ⊢ {𝐶, 𝐵} = {𝐵, 𝐶} | |
4 | 1, 2, 3 | 3eqtr4g 2175 | 1 ⊢ (𝐴 = 𝐵 → {𝐶, 𝐴} = {𝐶, 𝐵}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1316 {cpr 3498 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-v 2662 df-un 3045 df-sn 3503 df-pr 3504 |
This theorem is referenced by: preq12 3572 preq2i 3574 preq2d 3577 tpeq2 3580 preq12bg 3670 opeq2 3676 uniprg 3721 intprg 3774 prexg 4103 opth 4129 opeqsn 4144 relop 4659 funopg 5127 pr2ne 7016 hashprg 10522 bj-prexg 13036 |
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