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Mirrors > Home > MPE Home > Th. List > preq2i | Structured version Visualization version GIF version |
Description: Equality inference for unordered pairs. (Contributed by NM, 19-Oct-2012.) |
Ref | Expression |
---|---|
preq1i.1 | ⊢ 𝐴 = 𝐵 |
Ref | Expression |
---|---|
preq2i | ⊢ {𝐶, 𝐴} = {𝐶, 𝐵} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | preq1i.1 | . 2 ⊢ 𝐴 = 𝐵 | |
2 | preq2 4269 | . 2 ⊢ (𝐴 = 𝐵 → {𝐶, 𝐴} = {𝐶, 𝐵}) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ {𝐶, 𝐴} = {𝐶, 𝐵} |
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: opid 4421 funopg 5922 df2o2 7574 fzprval 12401 fz0to3un2pr 12441 fz0to4untppr 12442 fzo13pr 12552 fzo0to2pr 12553 fzo0to42pr 12555 bpoly3 14789 prmreclem2 15621 2strstr1 15986 mgmnsgrpex 17418 sgrpnmndex 17419 m2detleiblem2 20434 txindis 21437 iblcnlem1 23554 axlowdimlem4 25825 setsvtx 25927 uhgrwkspthlem2 26650 opidORIG 34109 opidg 41297 31prm 41512 nnsum3primes4 41676 nnsum3primesgbe 41680 |
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