| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| 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 4734 | . 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 1540 {cpr 4628 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-un 3956 df-sn 4627 df-pr 4629 |
| This theorem is referenced by: opidg 4892 funopg 6600 df2o2 8515 fz12pr 13621 fz0to3un2pr 13669 fz0to4untppr 13670 fzo13pr 13788 fzo0to2pr 13789 fz01pr 13790 fzo0to42pr 13792 bpoly3 16094 prmreclem2 16955 2strstr1OLD 17271 mgmnsgrpex 18944 sgrpnmndex 18945 m2detleiblem2 22634 txindis 23642 setsvtx 29052 uhgrwkspthlem2 29774 31prm 47584 nnsum3primes4 47775 nnsum3primesgbe 47779 |
| Copyright terms: Public domain | W3C validator |