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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 5150 | . 2 ⊢ (𝐴𝑅𝐵 ↔ 𝐴𝑅𝐶) | 
| 4 | 1, 3 | mpbi 230 | 1 ⊢ 𝐴𝑅𝐶 | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1539 class class class wbr 5142 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-br 5143 | 
| This theorem is referenced by: breqtrri 5169 3brtr3i 5171 supsrlem 11152 0lt1 11786 le9lt10 12762 9lt10 12866 hashunlei 14465 sqrt2gt1lt2 15314 trireciplem 15899 cos1bnd 16224 cos2bnd 16225 cos01gt0 16228 sin4lt0 16232 rpnnen2lem3 16253 z4even 16410 gcdaddmlem 16562 dec2dvds 17102 abvtrivd 20834 sincos4thpi 26556 log2cnv 26988 log2ublem2 26991 log2ublem3 26992 log2le1 26994 birthday 26998 harmonicbnd3 27052 lgam1 27108 basellem7 27131 ppiublem1 27247 ppiub 27249 bposlem4 27332 bposlem5 27333 bposlem9 27337 lgsdir2lem2 27371 lgsdir2lem3 27372 0reno 28430 ex-fl 30467 siilem1 30871 normlem5 31134 normlem6 31135 norm-ii-i 31157 norm3adifii 31168 cmm2i 31627 mayetes3i 31749 nmopcoadji 32121 mdoc2i 32446 dmdoc2i 32448 dp2lt10 32867 dp2ltsuc 32869 dplti 32888 sqsscirc1 33908 ballotlem1c 34511 hgt750lem 34667 problem5 35675 circum 35680 bj-pinftyccb 37223 bj-minftyccb 37227 poimirlem25 37653 cntotbnd 37804 3lexlogpow5ineq1 42056 3lexlogpow5ineq2 42057 aks4d1p1p2 42072 aks4d1p1p7 42076 posbezout 42102 aks6d1c7lem1 42182 jm2.23 43013 tr3dom 43546 halffl 45313 wallispi 46090 stirlinglem1 46094 fouriersw 46251 | 
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