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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 5053 | . 2 ⊢ 𝐶𝑅𝐵 |
4 | 3brtr3.3 | . 2 ⊢ 𝐵 = 𝐷 | |
5 | 3, 4 | breqtri 5055 | 1 ⊢ 𝐶𝑅𝐷 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1538 class class class wbr 5030 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-v 3443 df-un 3886 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 |
This theorem is referenced by: supsrlem 10522 ef01bndlem 15529 pige3ALT 25112 log2ublem1 25532 log2ub 25535 ppiublem1 25786 logfacrlim2 25810 chebbnd1 26056 nmoptri2i 29882 dpmul4 30616 problem5 33025 fouriersw 42873 |
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