Theorem eqbrtrrdi 3976

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 2146	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
3		eqbrtrrdi.2	. 2 ⊢ 𝐵𝑅𝐶
4	2, 3	eqbrtrdi 3975	1 ⊢ (𝜑 → 𝐴𝑅𝐶)

Syntax hints:   → wi 4   = wceq 1332   class class class wbr 3937

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-un 3080  df-sn 3538  df-pr 3539  df-op 3541  df-br 3938

This theorem is referenced by: (None)