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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 4864 | . 2 ⊢ 𝐶𝑅𝐵 |
4 | 3brtr4.3 | . 2 ⊢ 𝐷 = 𝐵 | |
5 | 3, 4 | breqtrri 4870 | 1 ⊢ 𝐶𝑅𝐷 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1653 class class class wbr 4843 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-rab 3098 df-v 3387 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-nul 4116 df-if 4278 df-sn 4369 df-pr 4371 df-op 4375 df-br 4844 |
This theorem is referenced by: 1lt2nq 10083 0lt1sr 10204 declt 11812 decltc 11813 decle 11818 fzennn 13022 faclbnd4lem1 13333 fsumabs 14871 ovolfiniun 23609 log2ublem3 25027 log2ub 25028 emgt0 25085 bclbnd 25357 bposlem8 25368 baseltedgf 26229 nmblolbii 28179 normlem6 28497 norm-ii-i 28519 nmbdoplbi 29408 dp2lt 30109 dp2ltsuc 30110 dp2ltc 30111 dplt 30128 dpltc 30131 dpmul4 30138 hgt750lemd 31246 hgt750lem 31249 supxrltinfxr 40420 nnsum4primesevenALTV 42471 |
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