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| Mirrors > Home > MPE Home > Th. List > eqbrtrri | Structured version Visualization version GIF version | ||
| Description: Substitution of equal classes into a binary relation. (Contributed by NM, 1-Aug-1999.) |
| Ref | Expression |
|---|---|
| eqbrtrr.1 | ⊢ 𝐴 = 𝐵 |
| eqbrtrr.2 | ⊢ 𝐴𝑅𝐶 |
| Ref | Expression |
|---|---|
| eqbrtrri | ⊢ 𝐵𝑅𝐶 |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eqbrtrr.1 | . . 3 ⊢ 𝐴 = 𝐵 | |
| 2 | 1 | eqcomi 2746 | . 2 ⊢ 𝐵 = 𝐴 |
| 3 | eqbrtrr.2 | . 2 ⊢ 𝐴𝑅𝐶 | |
| 4 | 2, 3 | eqbrtri 5164 | 1 ⊢ 𝐵𝑅𝐶 |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 class class class wbr 5143 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 |
| This theorem is referenced by: 3brtr3i 5172 expnass 14247 faclbnd4lem1 14332 sqrt2gt1lt2 15313 cos1bnd 16223 cos2bnd 16224 2strstr1OLD 17271 prdsvalstr 17497 ovolre 25560 pigt3 26560 pige3ALT 26562 atan1 26971 log2ublem1 26989 sqrtlim 27016 bposlem8 27335 chebbnd1 27516 nohalf 28407 norm-ii-i 31156 nmopadji 32109 unierri 32123 chnub 33002 ballotlem2 34491 hgt750lemd 34663 hgt750lem 34666 stoweidlem26 46041 wallispilem5 46084 |
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