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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 5171 | . 2 ⊢ 𝐶𝑅𝐵 |
4 | 3brtr3.3 | . 2 ⊢ 𝐵 = 𝐷 | |
5 | 3, 4 | breqtri 5173 | 1 ⊢ 𝐶𝑅𝐷 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 class class class wbr 5148 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 |
This theorem is referenced by: supsrlem 11149 ef01bndlem 16217 pige3ALT 26577 log2ublem1 27004 log2ub 27007 ppiublem1 27261 logfacrlim2 27285 chebbnd1 27531 nmoptri2i 32128 dpmul4 32881 problem5 35654 fouriersw 46187 |
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