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Mirrors > Home > ILE Home > Th. List > 3brtr4i | GIF version |
Description: Substitution of equality into both sides of a binary relation. (Contributed by NM, 11-Aug-1999.) |
Ref | Expression |
---|---|
3brtr4.1 | ⊢ 𝐴𝑅𝐵 |
3brtr4.2 | ⊢ 𝐶 = 𝐴 |
3brtr4.3 | ⊢ 𝐷 = 𝐵 |
Ref | Expression |
---|---|
3brtr4i | ⊢ 𝐶𝑅𝐷 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3brtr4.2 | . . 3 ⊢ 𝐶 = 𝐴 | |
2 | 3brtr4.1 | . . 3 ⊢ 𝐴𝑅𝐵 | |
3 | 1, 2 | eqbrtri 3870 | . 2 ⊢ 𝐶𝑅𝐵 |
4 | 3brtr4.3 | . 2 ⊢ 𝐷 = 𝐵 | |
5 | 3, 4 | breqtrri 3876 | 1 ⊢ 𝐶𝑅𝐷 |
Colors of variables: wff set class |
Syntax hints: = wceq 1290 class class class wbr 3851 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 |
This theorem depends on definitions: df-bi 116 df-3an 927 df-tru 1293 df-nf 1396 df-sb 1694 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-v 2622 df-un 3004 df-sn 3456 df-pr 3457 df-op 3459 df-br 3852 |
This theorem is referenced by: 1lt2nq 7026 0lt1sr 7372 ax0lt1 7472 declt 8965 decltc 8966 decle 8971 frecfzennn 9894 fsumabs 10920 2strbasg 11656 2stropg 11657 |
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