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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 4270 | . 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → {𝐴, 𝐶} = {𝐵, 𝐷}) | |
4 | 1, 2, 3 | mp2an 708 | 1 ⊢ {𝐴, 𝐶} = {𝐵, 𝐷} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1483 {cpr 4179 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-v 3202 df-un 3579 df-sn 4178 df-pr 4180 |
This theorem is referenced by: grpbasex 15994 grpplusgx 15995 indistpsx 20814 lgsdir2lem5 25054 wlk2v2elem2 27016 tgrpset 36033 zlmodzxzadd 42136 zlmodzxzequa 42285 zlmodzxzequap 42288 |
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