Theorem eqbrtrrdi 5114

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

Hypotheses

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

Assertion

Ref	Expression
eqbrtrrdi	⊢ (𝜑 → 𝐴𝑅𝐶)

Proof of Theorem eqbrtrrdi

Step	Hyp	Ref	Expression
1		eqbrtrrdi.1	. . 3 ⊢ (𝜑 → 𝐵 = 𝐴)
2	1	eqcomd 2744	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
3		eqbrtrrdi.2	. 2 ⊢ 𝐵𝑅𝐶
4	2, 3	eqbrtrdi 5113	1 ⊢ (𝜑 → 𝐴𝑅𝐶)

Syntax hints:   → wi 4   = wceq 1539   class class class wbr 5074

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075

This theorem is referenced by:  grur1  10576  t1connperf  22587  basellem9  26238  sqff1o  26331  ballotlemic  32473  ballotlem1c  32474  pibt2  35588