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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 4663 | . 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 1533 {cpr 4562 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-v 3496 df-un 3940 df-sn 4561 df-pr 4563 |
This theorem is referenced by: opidg 4815 funopg 6383 df2o2 8112 fz12pr 12958 fz0to3un2pr 13003 fz0to4untppr 13004 fzo13pr 13115 fzo0to2pr 13116 fzo0to42pr 13118 bpoly3 15406 prmreclem2 16247 2strstr1 16599 mgmnsgrpex 18090 sgrpnmndex 18091 m2detleiblem2 21231 txindis 22236 setsvtx 26814 uhgrwkspthlem2 27529 31prm 43754 nnsum3primes4 43947 nnsum3primesgbe 43951 |
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