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Mirrors > Home > MPE Home > Th. List > preq1i | Structured version Visualization version GIF version |
Description: Equality inference for unordered pairs. (Contributed by NM, 19-Oct-2012.) |
Ref | Expression |
---|---|
preq1i.1 | ⊢ 𝐴 = 𝐵 |
Ref | Expression |
---|---|
preq1i | ⊢ {𝐴, 𝐶} = {𝐵, 𝐶} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | preq1i.1 | . 2 ⊢ 𝐴 = 𝐵 | |
2 | preq1 4212 | . 2 ⊢ (𝐴 = 𝐵 → {𝐴, 𝐶} = {𝐵, 𝐶}) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ {𝐴, 𝐶} = {𝐵, 𝐶} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 {cpr 4127 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-v 3175 df-un 3545 df-sn 4126 df-pr 4128 |
This theorem is referenced by: funopg 5836 frgraunss 26522 n4cyclfrgra 26545 disjdifprg2 28771 frcond1 41438 n4cyclfrgr 41461 |
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