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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 4701 | . 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → {𝐴, 𝐶} = {𝐵, 𝐷}) | |
4 | 1, 2, 3 | mp2an 691 | 1 ⊢ {𝐴, 𝐶} = {𝐵, 𝐷} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 {cpr 4593 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2708 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-v 3450 df-un 3920 df-sn 4592 df-pr 4594 |
This theorem is referenced by: grpbasex 17179 grpplusgx 17180 indistpsx 22376 lgsdir2lem5 26693 wlk2v2elem2 29142 tgrpset 39237 zlmodzxzadd 46508 zlmodzxzequa 46651 zlmodzxzequap 46654 |
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