Theorem syl6eqbrr 4965

Description: A chained equality inference for a binary relation. (Contributed by NM, 4-Jan-2006.)

Hypotheses

Ref	Expression
syl6eqbrr.1	⊢ (𝜑 → 𝐵 = 𝐴)
syl6eqbrr.2	⊢ 𝐵𝑅𝐶

Assertion

Ref	Expression
syl6eqbrr	⊢ (𝜑 → 𝐴𝑅𝐶)

Proof of Theorem syl6eqbrr

Step	Hyp	Ref	Expression
1		syl6eqbrr.1	. . 3 ⊢ (𝜑 → 𝐵 = 𝐴)
2	1	eqcomd 2777	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
3		syl6eqbrr.2	. 2 ⊢ 𝐵𝑅𝐶
4	2, 3	syl6eqbr 4964	1 ⊢ (𝜑 → 𝐴𝑅𝐶)

Syntax hints:   → wi 4   = wceq 1508   class class class wbr 4925

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-ext 2743

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-rab 3090  df-v 3410  df-dif 3825  df-un 3827  df-in 3829  df-ss 3836  df-nul 4173  df-if 4345  df-sn 4436  df-pr 4438  df-op 4442  df-br 4926

This theorem is referenced by:  grur1  10038  t1connperf  21763  basellem9  25383  sqff1o  25476  ballotlemic  31442  ballotlem1c  31443  pibt2  34176