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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 7257 poxp 8116 poxp2 8131 poxp3 8138 unxpdomlem1 9252 unxpdomlem2 9253 aceq0 10115 zfac 10457 axrepndlem1 10589 zfcndac 10616 injresinj 13757 fsum2dlem 15720 ramub1lem2 16964 ramcl 16966 symgextf1 19330 mamulid 22163 mamurid 22164 mdetdiagid 22322 mdetunilem9 22342 alexsubALTlem3 23773 ptcmplem2 23777 dscmet 24301 dyadmbllem 25340 opnmbllem 25342 isppw2 26843 2sqreulem1 27173 2sqreunnlem1 27176 frgr2wwlk1 29837 disji2f 32063 disjif2 32067 bj-ssblem1 35834 bj-ssblem2 35835 cbveud 36556 wl-naevhba1v 36692 wl-equsb3 36724 mblfinlem1 36828 bfp 36995 dveeq1-o 38108 dveeq1-o16 38109 axc11n-16 38111 ax12eq 38114 fsuppind 41464 fphpd 41856 ax6e2nd 43621 ax6e2ndVD 43971 ax6e2ndALT 43993 disjinfi 44190 iundjiun 45475 hspdifhsp 45631 hspmbl 45644 2reu8i 46120 2reuimp0 46121 ichexmpl1 46436 lcoss 47205 |
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