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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 4672 | . 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 1537 {cpr 4571 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-v 3498 df-un 3943 df-sn 4570 df-pr 4572 |
This theorem is referenced by: opidg 4824 funopg 6391 df2o2 8120 fz12pr 12967 fz0to3un2pr 13012 fz0to4untppr 13013 fzo13pr 13124 fzo0to2pr 13125 fzo0to42pr 13127 bpoly3 15414 prmreclem2 16255 2strstr1 16607 mgmnsgrpex 18098 sgrpnmndex 18099 m2detleiblem2 21239 txindis 22244 setsvtx 26822 uhgrwkspthlem2 27537 31prm 43767 nnsum3primes4 43960 nnsum3primesgbe 43964 |
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