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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 4676 | . 2 ⊢ 𝐶𝑅𝐵 |
4 | 3brtr3.3 | . 2 ⊢ 𝐵 = 𝐷 | |
5 | 3, 4 | breqtri 4678 | 1 ⊢ 𝐶𝑅𝐷 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1483 class class class wbr 4653 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 |
This theorem is referenced by: supsrlem 9932 ef01bndlem 14914 pige3 24269 log2ublem1 24673 log2ub 24676 ppiublem1 24927 logfacrlim2 24951 chebbnd1 25161 nmoptri2i 28958 dpmul4 29622 problem5 31563 fouriersw 40448 |
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