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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 3684 | . 2 ⊢ (𝐴 = 𝐵 → {𝐴, 𝐶} = {𝐵, 𝐶}) | |
2 | prcom 3683 | . 2 ⊢ {𝐶, 𝐴} = {𝐴, 𝐶} | |
3 | prcom 3683 | . 2 ⊢ {𝐶, 𝐵} = {𝐵, 𝐶} | |
4 | 1, 2, 3 | 3eqtr4g 2247 | 1 ⊢ (𝐴 = 𝐵 → {𝐶, 𝐴} = {𝐶, 𝐵}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1364 {cpr 3608 |
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-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-v 2754 df-un 3148 df-sn 3613 df-pr 3614 |
This theorem is referenced by: preq12 3686 preq2i 3688 preq2d 3691 tpeq2 3694 preq12bg 3788 opeq2 3794 uniprg 3839 intprg 3892 prexg 4229 opth 4255 opeqsn 4270 relop 4795 funopg 5269 pr2ne 7220 hashprg 10819 bj-prexg 15116 |
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