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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 4742 | . 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → {𝐴, 𝐶} = {𝐵, 𝐷}) | |
4 | 1, 2, 3 | mp2an 690 | 1 ⊢ {𝐴, 𝐶} = {𝐵, 𝐷} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 {cpr 4632 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2698 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2705 df-cleq 2719 df-clel 2805 df-v 3473 df-un 3952 df-sn 4631 df-pr 4633 |
This theorem is referenced by: grpbasex 17277 grpplusgx 17278 indistpsx 22931 lgsdir2lem5 27280 wlk2v2elem2 29984 tgrpset 40222 zlmodzxzadd 47473 zlmodzxzequa 47615 zlmodzxzequap 47618 |
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