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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 4489 | . 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → {𝐴, 𝐶} = {𝐵, 𝐷}) | |
4 | 1, 2, 3 | mp2an 685 | 1 ⊢ {𝐴, 𝐶} = {𝐵, 𝐷} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1658 {cpr 4400 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-ext 2804 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-v 3417 df-un 3804 df-sn 4399 df-pr 4401 |
This theorem is referenced by: grpbasex 16354 grpplusgx 16355 indistpsx 21186 lgsdir2lem5 25468 wlk2v2elem2 27533 tgrpset 36821 zlmodzxzadd 42984 zlmodzxzequa 43133 zlmodzxzequap 43136 |
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