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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 2012 | . 2 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝑧 → 𝑦 = 𝑧)) | |
2 | equtr 2017 | . 2 ⊢ (𝑥 = 𝑦 → (𝑦 = 𝑧 → 𝑥 = 𝑧)) | |
3 | 1, 2 | impbid 212 | 1 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝑧 ↔ 𝑦 = 𝑧)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 |
This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1776 |
This theorem is referenced by: equvinv 2025 equvelv 2027 spaev 2049 sbjust 2060 equsb3 2100 cbvsbvf 2363 drsb1 2497 mo4 2563 sb8eulem 2595 cbvmovw 2599 cbvmow 2600 axextg 2707 reu6 3734 reu7 3740 reu8nf 3885 disjxun 5145 solin 5622 cbviotaw 6522 cbviota 6524 dff13f 7275 poxp 8151 poxp2 8166 poxp3 8173 unxpdomlem1 9283 unxpdomlem2 9284 aceq0 10155 zfac 10497 axrepndlem1 10629 zfcndac 10656 injresinj 13823 fsum2dlem 15802 ramub1lem2 17060 ramcl 17062 symgextf1 19453 mamulid 22462 mamurid 22463 mdetdiagid 22621 mdetunilem9 22641 alexsubALTlem3 24072 ptcmplem2 24076 dscmet 24600 dyadmbllem 25647 opnmbllem 25649 isppw2 27172 2sqreulem1 27504 2sqreunnlem1 27507 frgr2wwlk1 30357 disji2f 32596 disjif2 32600 cbvmodavw 36232 cbvsbdavw 36236 cbvsbdavw2 36237 bj-ssblem1 36636 bj-ssblem2 36637 cbveud 37354 wl-naevhba1v 37500 wl-equsb3 37536 mblfinlem1 37643 bfp 37810 dveeq1-o 38916 dveeq1-o16 38917 axc11n-16 38919 ax12eq 38922 aks6d1c6lem3 42153 aks6d1c7 42165 fsuppind 42576 eu6w 42662 fphpd 42803 ax6e2nd 44555 ax6e2ndVD 44905 ax6e2ndALT 44927 disjinfi 45134 iundjiun 46415 hspdifhsp 46571 hspmbl 46584 2reu8i 47062 2reuimp0 47063 ichexmpl1 47393 lcoss 48281 |
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