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Mirrors > Home > ILE Home > Th. List > breqtrri | GIF version |
Description: Substitution of equal classes into a binary relation. (Contributed by NM, 5-Aug-1993.) |
Ref | Expression |
---|---|
breqtrr.1 | ⊢ 𝐴𝑅𝐵 |
breqtrr.2 | ⊢ 𝐶 = 𝐵 |
Ref | Expression |
---|---|
breqtrri | ⊢ 𝐴𝑅𝐶 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breqtrr.1 | . 2 ⊢ 𝐴𝑅𝐵 | |
2 | breqtrr.2 | . . 3 ⊢ 𝐶 = 𝐵 | |
3 | 2 | eqcomi 2181 | . 2 ⊢ 𝐵 = 𝐶 |
4 | 1, 3 | breqtri 4027 | 1 ⊢ 𝐴𝑅𝐶 |
Colors of variables: wff set class |
Syntax hints: = wceq 1353 class class class wbr 4002 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-v 2739 df-un 3133 df-sn 3598 df-pr 3599 df-op 3601 df-br 4003 |
This theorem is referenced by: 3brtr4i 4032 ensn1 6793 pw1dom2 7223 0lt1sr 7761 0le2 9005 2pos 9006 3pos 9009 4pos 9012 5pos 9015 6pos 9016 7pos 9017 8pos 9018 9pos 9019 1lt2 9084 2lt3 9085 3lt4 9087 4lt5 9090 5lt6 9094 6lt7 9099 7lt8 9105 8lt9 9112 nn0le2xi 9222 numltc 9405 declti 9417 sqge0i 10601 faclbnd2 10715 ege2le3 11672 cos2bnd 11761 3dvdsdec 11862 n2dvdsm1 11910 n2dvds3 11912 pockthi 12348 dveflem 14058 tangtx 14130 lgsdir2lem2 14301 ex-fl 14337 |
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