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| Mirrors > Home > MPE Home > Th. List > preq12i | Structured version Visualization version GIF version | ||
| Description: Equality inference for unordered pairs. (Contributed by NM, 19-Oct-2012.) |
| Ref | Expression |
|---|---|
| preq1i.1 | ⊢ 𝐴 = 𝐵 |
| preq12i.2 | ⊢ 𝐶 = 𝐷 |
| Ref | Expression |
|---|---|
| preq12i | ⊢ {𝐴, 𝐶} = {𝐵, 𝐷} |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | preq1i.1 | . 2 ⊢ 𝐴 = 𝐵 | |
| 2 | preq12i.2 | . 2 ⊢ 𝐶 = 𝐷 | |
| 3 | preq12 4735 | . 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → {𝐴, 𝐶} = {𝐵, 𝐷}) | |
| 4 | 1, 2, 3 | mp2an 692 | 1 ⊢ {𝐴, 𝐶} = {𝐵, 𝐷} |
| 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: grpbasex 17335 grpplusgx 17336 indistpsx 23017 lgsdir2lem5 27373 negs1s 28059 wlk2v2elem2 30175 tgrpset 40747 stgr0 47927 stgr1 47928 zlmodzxzadd 48274 zlmodzxzequa 48413 zlmodzxzequap 48416 |
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