Theorem syl6eqbr 3912

Description: A chained equality inference for a binary relation. (Contributed by NM, 12-Oct-1999.)

Hypotheses

Ref	Expression
syl6eqbr.1	⊢ (𝜑 → 𝐴 = 𝐵)
syl6eqbr.2	⊢ 𝐵𝑅𝐶

Assertion

Ref	Expression
syl6eqbr	⊢ (𝜑 → 𝐴𝑅𝐶)

Proof of Theorem syl6eqbr

Step	Hyp	Ref	Expression
1		syl6eqbr.2	. 2 ⊢ 𝐵𝑅𝐶
2		syl6eqbr.1	. . 3 ⊢ (𝜑 → 𝐴 = 𝐵)
3	2	breq1d 3885	. 2 ⊢ (𝜑 → (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐶))
4	1, 3	mpbiri 167	1 ⊢ (𝜑 → 𝐴𝑅𝐶)

Syntax hints:   → wi 4   = wceq 1299   class class class wbr 3875

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 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082

This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-v 2643  df-un 3025  df-sn 3480  df-pr 3481  df-op 3483  df-br 3876

This theorem is referenced by:  syl6eqbrr  3913  pm54.43  6957  nn0ledivnn  9395  xltnegi  9459  leexp1a  10189  facwordi  10327  faclbnd3  10330  resqrexlemlo  10625  efap0  11181  dvds1  11346  en1top  12028