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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 4896 | . 2 ⊢ 𝐶𝑅𝐵 |
4 | 3brtr4.3 | . 2 ⊢ 𝐷 = 𝐵 | |
5 | 3, 4 | breqtrri 4902 | 1 ⊢ 𝐶𝑅𝐷 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1656 class class class wbr 4875 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-rab 3126 df-v 3416 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-sn 4400 df-pr 4402 df-op 4406 df-br 4876 |
This theorem is referenced by: 1lt2nq 10117 0lt1sr 10239 declt 11857 decltc 11858 decle 11863 fzennn 13069 faclbnd4lem1 13380 fsumabs 14914 ovolfiniun 23674 log2ublem3 25095 log2ub 25096 emgt0 25153 bclbnd 25425 bposlem8 25436 baseltedgf 26299 nmblolbii 28205 normlem6 28523 norm-ii-i 28545 nmbdoplbi 29434 dp2lt 30134 dp2ltsuc 30135 dp2ltc 30136 dplt 30153 dpltc 30156 dpmul4 30163 hgt750lemd 31271 hgt750lem 31274 supxrltinfxr 40470 nnsum4primesevenALTV 42537 |
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