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Mirrors > Home > MPE Home > Th. List > preq12 | Structured version Visualization version GIF version |
Description: Equality theorem for unordered pairs. (Contributed by NM, 19-Oct-2012.) |
Ref | Expression |
---|---|
preq12 | ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → {𝐴, 𝐵} = {𝐶, 𝐷}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | preq1 4662 | . 2 ⊢ (𝐴 = 𝐶 → {𝐴, 𝐵} = {𝐶, 𝐵}) | |
2 | preq2 4663 | . 2 ⊢ (𝐵 = 𝐷 → {𝐶, 𝐵} = {𝐶, 𝐷}) | |
3 | 1, 2 | sylan9eq 2876 | 1 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → {𝐴, 𝐵} = {𝐶, 𝐷}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = 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: preq12i 4667 preq12d 4670 ssprsseq 4751 preq12b 4774 prnebg 4779 preq12nebg 4786 opthprneg 4788 snex 5323 relop 5715 opthreg 9075 hashle2pr 13829 wwlktovfo 14316 joinval 17609 meetval 17623 ipole 17762 sylow1 18722 frgpuplem 18892 uspgr2wlkeq 27421 wlkres 27446 wlkp1lem8 27456 usgr2pthlem 27538 2wlkdlem10 27708 1wlkdlem4 27913 3wlkdlem6 27938 3wlkdlem10 27942 pfxwlk 32365 oppr 43259 imarnf1pr 43475 elsprel 43631 sprsymrelf1lem 43647 sprsymrelf 43651 paireqne 43667 sbcpr 43677 isomuspgrlem2b 43988 isomuspgrlem2d 43990 |
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