Theorem breqtri 4084

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

Hypotheses

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

Assertion

Ref	Expression
breqtri	⊢ 𝐴𝑅𝐶

Proof of Theorem breqtri

Step	Hyp	Ref	Expression
1		breqtr.1	. 2 ⊢ 𝐴𝑅𝐵
2		breqtr.2	. . 3 ⊢ 𝐵 = 𝐶
3	2	breq2i 4067	. 2 ⊢ (𝐴𝑅𝐵 ↔ 𝐴𝑅𝐶)
4	1, 3	mpbi 145	1 ⊢ 𝐴𝑅𝐶

Syntax hints:   = wceq 1373   class class class wbr 4059

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-un 3178  df-sn 3649  df-pr 3650  df-op 3652  df-br 4060

This theorem is referenced by:  breqtrri  4086  3brtr3i  4088  le9lt10  9565  9lt10  9669  sqrt2gt1lt2  11475  trireciplem  11926  cos1bnd  12185  cos2bnd  12186  cos01gt0  12189  sin4lt0  12193  z4even  12342  dec2dvds  12849  coseq00topi  15422  sincos4thpi  15427  lgsdir2lem2  15621  lgsdir2lem3  15622  ex-fl  15861