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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 5074 | 1 ⊢ 𝐵𝑅𝐶 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1543 class class class wbr 5053 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2071 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3070 df-v 3410 df-dif 3869 df-un 3871 df-nul 4238 df-if 4440 df-sn 4542 df-pr 4544 df-op 4548 df-br 5054 |
This theorem is referenced by: 3brtr3i 5082 expnass 13776 faclbnd4lem1 13859 sqrt2gt1lt2 14838 cos1bnd 15748 cos2bnd 15749 2strstr1 16781 prdsvalstr 16957 ovolre 24422 pigt3 25407 pige3ALT 25409 atan1 25811 log2ublem1 25829 sqrtlim 25855 bposlem8 26172 chebbnd1 26353 norm-ii-i 29218 nmopadji 30171 unierri 30185 ballotlem2 32167 hgt750lemd 32340 hgt750lem 32343 stoweidlem26 43242 wallispilem5 43285 |
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