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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 4699 | . 2 ⊢ (𝐴 = 𝐶 → {𝐴, 𝐵} = {𝐶, 𝐵}) | |
2 | preq2 4700 | . 2 ⊢ (𝐵 = 𝐷 → {𝐶, 𝐵} = {𝐶, 𝐷}) | |
3 | 1, 2 | sylan9eq 2797 | 1 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → {𝐴, 𝐵} = {𝐶, 𝐷}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 {cpr 4593 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2708 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-v 3450 df-un 3920 df-sn 4592 df-pr 4594 |
This theorem is referenced by: preq12i 4704 preq12d 4707 ssprsseq 4790 preq12b 4813 prnebg 4818 preq12nebg 4825 opthprneg 4827 relop 5811 opthreg 9561 hashle2pr 14383 wwlktovfo 14854 joinval 18273 meetval 18287 ipole 18430 sylow1 19392 frgpuplem 19561 uspgr2wlkeq 28636 wlkres 28660 wlkp1lem8 28670 usgr2pthlem 28753 2wlkdlem10 28922 1wlkdlem4 29126 3wlkdlem6 29151 3wlkdlem10 29155 pfxwlk 33757 oppr 45338 imarnf1pr 45588 elsprel 45741 sprsymrelf1lem 45757 sprsymrelf 45761 paireqne 45777 sbcpr 45787 isomuspgrlem2b 46095 isomuspgrlem2d 46097 |
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