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Mirrors > Home > MPE Home > Th. List > equequ1 | Structured version Visualization version GIF version |
Description: An equivalence law for equality. (Contributed by NM, 1-Aug-1993.) (Proof shortened by Wolf Lammen, 10-Dec-2017.) |
Ref | Expression |
---|---|
equequ1 | ⊢ (𝑥 = 𝑦 → (𝑥 = 𝑧 ↔ 𝑦 = 𝑧)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax7 2019 | . 2 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝑧 → 𝑦 = 𝑧)) | |
2 | equtr 2024 | . 2 ⊢ (𝑥 = 𝑦 → (𝑦 = 𝑧 → 𝑥 = 𝑧)) | |
3 | 1, 2 | impbid 211 | 1 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝑧 ↔ 𝑦 = 𝑧)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 |
This theorem depends on definitions: df-bi 206 df-an 397 df-ex 1782 |
This theorem is referenced by: equvinv 2032 equvelv 2034 spaev 2055 sbjust 2066 equsb3 2101 cbvsbvf 2360 drsb1 2494 mo4 2560 sb8eulem 2592 cbvmovw 2596 cbvmow 2597 axextg 2705 reu6 3722 reu7 3728 reu8nf 3871 disjxun 5146 solin 5613 cbviotaw 6502 cbviota 6505 dff13f 7254 poxp 8113 poxp2 8128 poxp3 8135 unxpdomlem1 9249 unxpdomlem2 9250 aceq0 10112 zfac 10454 axrepndlem1 10586 zfcndac 10613 injresinj 13752 fsum2dlem 15715 ramub1lem2 16959 ramcl 16961 symgextf1 19288 mamulid 21942 mamurid 21943 mdetdiagid 22101 mdetunilem9 22121 alexsubALTlem3 23552 ptcmplem2 23556 dscmet 24080 dyadmbllem 25115 opnmbllem 25117 isppw2 26616 2sqreulem1 26946 2sqreunnlem1 26949 frgr2wwlk1 29579 disji2f 31803 disjif2 31807 bj-ssblem1 35526 bj-ssblem2 35527 cbveud 36248 wl-naevhba1v 36384 wl-equsb3 36416 mblfinlem1 36520 bfp 36687 dveeq1-o 37800 dveeq1-o16 37801 axc11n-16 37803 ax12eq 37806 fsuppind 41164 fphpd 41544 ax6e2nd 43309 ax6e2ndVD 43659 ax6e2ndALT 43681 disjinfi 43881 iundjiun 45166 hspdifhsp 45322 hspmbl 45335 2reu8i 45811 2reuimp0 45812 ichexmpl1 46127 lcoss 47107 |
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