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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 4733 | . 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 1540 {cpr 4628 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-un 3956 df-sn 4627 df-pr 4629 | 
| This theorem is referenced by: funopg 6600 frcond1 30285 n4cyclfrgr 30310 disjdifprg2 32589 | 
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