Theorem breqtrri 3789

Description: Substitution of equal classes into a binary relation. (Contributed by NM, 5-Aug-1993.)

Hypotheses

Ref	Expression
breqtrr.1	⊢ 𝐴𝑅𝐵
breqtrr.2	⊢ 𝐶 = 𝐵

Assertion

Ref	Expression
breqtrri	⊢ 𝐴𝑅𝐶

Proof of Theorem breqtrri

Step	Hyp	Ref	Expression
1		breqtrr.1	. 2 ⊢ 𝐴𝑅𝐵
2		breqtrr.2	. . 3 ⊢ 𝐶 = 𝐵
3	2	eqcomi 2044	. 2 ⊢ 𝐵 = 𝐶
4	1, 3	breqtri 3787	1 ⊢ 𝐴𝑅𝐶

Syntax hints:   = wceq 1243   class class class wbr 3764

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765

This theorem is referenced by:  3brtr4i  3792  ensn1  6276  0lt1sr  6850  0le2  8006  2pos  8007  3pos  8010  4pos  8013  5pos  8016  6pos  8017  7pos  8018  8pos  8019  9pos  8020  10pos  8021  1lt2  8086  2lt3  8087  3lt4  8089  4lt5  8092  5lt6  8096  6lt7  8101  7lt8  8107  8lt9  8114  9lt10  8122  nn0le2xi  8232  numltc  8387  declti  8392  sqge0i  9340  ex-fl  9895