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Mirrors > Home > MPE Home > Th. List > preq1d | Structured version Visualization version GIF version |
Description: Equality deduction for unordered pairs. (Contributed by NM, 19-Oct-2012.) |
Ref | Expression |
---|---|
preq1d.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
preq1d | ⊢ (𝜑 → {𝐴, 𝐶} = {𝐵, 𝐶}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | preq1d.1 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | preq1 4212 | . 2 ⊢ (𝐴 = 𝐵 → {𝐴, 𝐶} = {𝐵, 𝐶}) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝜑 → {𝐴, 𝐶} = {𝐵, 𝐶}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 {cpr 4127 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-v 3175 df-un 3545 df-sn 4126 df-pr 4128 |
This theorem is referenced by: propeqop 4895 opthwiener 4901 fprg 6327 fnpr2g 6379 dfac2 8836 symg2bas 17641 2pthoncl 26133 wwlknredwwlkn 26254 wwlkextprop 26272 clwwlkgt0 26299 clwlkisclwwlklem2fv1 26310 clwlkisclwwlklem2fv2 26311 clwlkisclwwlklem2a 26313 clwlkisclwwlklem0 26316 clwwisshclwwlem 26334 eupath2lem3 26506 frgraunss 26522 frgra1v 26525 frgra2v 26526 frgra3v 26529 n4cyclfrgra 26545 fprb 30916 wopprc 36615 crctcsh1wlkn0lem6 41018 wwlksnredwwlkn 41101 wwlksnextprop 41118 clwlkclwwlklem2fv1 41204 clwlkclwwlklem2fv2 41205 clwlkclwwlklem2a 41207 clwlkclwwlklem3 41210 clwwlks1loop 41215 clwwlksn1loop 41216 clwwisshclwwslem 41234 frcond1 41438 frgr1v 41441 nfrgr2v 41442 frgr3v 41445 n4cyclfrgr 41461 |
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