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Mirrors > Home > MPE Home > Th. List > breqtri | Structured version Visualization version GIF version |
Description: Substitution of equal classes into a binary relation. (Contributed by NM, 1-Aug-1999.) |
Ref | Expression |
---|---|
breqtr.1 | ⊢ 𝐴𝑅𝐵 |
breqtr.2 | ⊢ 𝐵 = 𝐶 |
Ref | Expression |
---|---|
breqtri | ⊢ 𝐴𝑅𝐶 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breqtr.1 | . 2 ⊢ 𝐴𝑅𝐵 | |
2 | breqtr.2 | . . 3 ⊢ 𝐵 = 𝐶 | |
3 | 2 | breq2i 5156 | . 2 ⊢ (𝐴𝑅𝐵 ↔ 𝐴𝑅𝐶) |
4 | 1, 3 | mpbi 230 | 1 ⊢ 𝐴𝑅𝐶 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 class class class wbr 5148 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 |
This theorem is referenced by: breqtrri 5175 3brtr3i 5177 supsrlem 11149 0lt1 11783 le9lt10 12758 9lt10 12862 hashunlei 14461 sqrt2gt1lt2 15310 trireciplem 15895 cos1bnd 16220 cos2bnd 16221 cos01gt0 16224 sin4lt0 16228 rpnnen2lem3 16249 z4even 16406 gcdaddmlem 16558 dec2dvds 17097 abvtrivd 20850 sincos4thpi 26570 log2cnv 27002 log2ublem2 27005 log2ublem3 27006 log2le1 27008 birthday 27012 harmonicbnd3 27066 lgam1 27122 basellem7 27145 ppiublem1 27261 ppiub 27263 bposlem4 27346 bposlem5 27347 bposlem9 27351 lgsdir2lem2 27385 lgsdir2lem3 27386 0reno 28444 ex-fl 30476 siilem1 30880 normlem5 31143 normlem6 31144 norm-ii-i 31166 norm3adifii 31177 cmm2i 31636 mayetes3i 31758 nmopcoadji 32130 mdoc2i 32455 dmdoc2i 32457 dp2lt10 32851 dp2ltsuc 32853 dplti 32872 sqsscirc1 33869 ballotlem1c 34489 hgt750lem 34645 problem5 35654 circum 35659 bj-pinftyccb 37204 bj-minftyccb 37208 poimirlem25 37632 cntotbnd 37783 3lexlogpow5ineq1 42036 3lexlogpow5ineq2 42037 aks4d1p1p2 42052 aks4d1p1p7 42056 posbezout 42082 aks6d1c7lem1 42162 jm2.23 42985 tr3dom 43518 halffl 45247 wallispi 46026 stirlinglem1 46030 fouriersw 46187 |
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