Theorem eqbrtrrdi 5145

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 2767	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
3		eqbrtrrdi.2	. 2 ⊢ 𝐵𝑅𝐶
4	2, 3	eqbrtrdi 5144	1 ⊢ (𝜑 → 𝐴𝑅𝐶)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   class class class wbr 5103

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104

This theorem is used by:  grur1  10898  t1connperf  23747  basellem9  27409  sqff1o  27502  ballotlemic  35132  ballotlem1c  35133  pibt2  38320