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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 5081 | . 2 ⊢ 𝐶𝑅𝐵 |
4 | 3brtr3.3 | . 2 ⊢ 𝐵 = 𝐷 | |
5 | 3, 4 | breqtri 5083 | 1 ⊢ 𝐶𝑅𝐷 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 class class class wbr 5058 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-br 5059 |
This theorem is referenced by: supsrlem 10527 ef01bndlem 15531 pige3ALT 25099 log2ublem1 25518 log2ub 25521 ppiublem1 25772 logfacrlim2 25796 chebbnd1 26042 nmoptri2i 29870 dpmul4 30585 problem5 32907 fouriersw 42510 |
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