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| Mirrors > Home > MPE Home > Th. List > 3brtr4i | Structured version Visualization version GIF version | ||
| Description: Substitution of equality into both sides of a binary relation. (Contributed by NM, 11-Aug-1999.) |
| Ref | Expression |
|---|---|
| 3brtr4.1 | ⊢ 𝐴𝑅𝐵 |
| 3brtr4.2 | ⊢ 𝐶 = 𝐴 |
| 3brtr4.3 | ⊢ 𝐷 = 𝐵 |
| Ref | Expression |
|---|---|
| 3brtr4i | ⊢ 𝐶𝑅𝐷 |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | 3brtr4.2 | . . 3 ⊢ 𝐶 = 𝐴 | |
| 2 | 3brtr4.1 | . . 3 ⊢ 𝐴𝑅𝐵 | |
| 3 | 1, 2 | eqbrtri 5164 | . 2 ⊢ 𝐶𝑅𝐵 |
| 4 | 3brtr4.3 | . 2 ⊢ 𝐷 = 𝐵 | |
| 5 | 3, 4 | breqtrri 5170 | 1 ⊢ 𝐶𝑅𝐷 |
| 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: 1lt2nq 11013 0lt1sr 11135 declt 12761 decltc 12762 decle 12767 fzennn 14009 faclbnd4lem1 14332 fsumabs 15837 basendxltplusgndx 17326 basendxlttsetndx 17399 basendxltplendx 17413 basendxltdsndx 17432 basendxltunifndx 17442 ovolfiniun 25536 log2ublem3 26991 log2ub 26992 bclbnd 27324 bposlem8 27335 basendxltedgfndx 29010 baseltedgfOLD 29011 nmblolbii 30818 normlem6 31134 norm-ii-i 31156 nmbdoplbi 32043 dp2lt 32867 dp2ltsuc 32868 dp2ltc 32869 dplt 32886 dpltc 32889 dpmul4 32896 hgt750lemd 34663 hgt750lem 34666 supxrltinfxr 45460 nnsum4primesevenALTV 47788 |
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