Theorem syl5breqr 3821

Description: B chained equality inference for a binary relation. (Contributed by NM, 24-Apr-2005.)

Hypotheses

Ref	Expression
syl5breqr.1	⊢ 𝐴𝑅𝐵
syl5breqr.2	⊢ (𝜑 → 𝐶 = 𝐵)

Assertion

Ref	Expression
syl5breqr	⊢ (𝜑 → 𝐴𝑅𝐶)

Proof of Theorem syl5breqr

Step	Hyp	Ref	Expression
1		syl5breqr.1	. 2 ⊢ 𝐴𝑅𝐵
2		syl5breqr.2	. . 3 ⊢ (𝜑 → 𝐶 = 𝐵)
3	2	eqcomd 2086	. 2 ⊢ (𝜑 → 𝐵 = 𝐶)
4	1, 3	syl5breq 3820	1 ⊢ (𝜑 → 𝐴𝑅𝐶)

Syntax hints:   → wi 4   = wceq 1284   class class class wbr 3785

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-un 2977  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786

This theorem is referenced by:  bernneq  9593