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Mirrors > Home > MPE Home > Th. List > 3brtr3i | 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 |
---|---|
3brtr3.1 | ⊢ 𝐴𝑅𝐵 |
3brtr3.2 | ⊢ 𝐴 = 𝐶 |
3brtr3.3 | ⊢ 𝐵 = 𝐷 |
Ref | Expression |
---|---|
3brtr3i | ⊢ 𝐶𝑅𝐷 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3brtr3.2 | . . 3 ⊢ 𝐴 = 𝐶 | |
2 | 3brtr3.1 | . . 3 ⊢ 𝐴𝑅𝐵 | |
3 | 1, 2 | eqbrtrri 5090 | . 2 ⊢ 𝐶𝑅𝐵 |
4 | 3brtr3.3 | . 2 ⊢ 𝐵 = 𝐷 | |
5 | 3, 4 | breqtri 5092 | 1 ⊢ 𝐶𝑅𝐷 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1543 class class class wbr 5067 |
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 2113 ax-9 2121 ax-ext 2709 |
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 2072 df-clab 2716 df-cleq 2730 df-clel 2817 df-rab 3071 df-v 3422 df-dif 3883 df-un 3885 df-nul 4252 df-if 4454 df-sn 4556 df-pr 4558 df-op 4562 df-br 5068 |
This theorem is referenced by: supsrlem 10749 ef01bndlem 15769 pige3ALT 25433 log2ublem1 25853 log2ub 25856 ppiublem1 26107 logfacrlim2 26131 chebbnd1 26377 nmoptri2i 30204 dpmul4 30932 problem5 33363 fouriersw 43475 |
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