Theorem syl6breqr 3883

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

Hypotheses

Ref	Expression
syl6breqr.1	⊢ (𝜑 → 𝐴𝑅𝐵)
syl6breqr.2	⊢ 𝐶 = 𝐵

Assertion

Ref	Expression
syl6breqr	⊢ (𝜑 → 𝐴𝑅𝐶)

Proof of Theorem syl6breqr

Step	Hyp	Ref	Expression
1		syl6breqr.1	. 2 ⊢ (𝜑 → 𝐴𝑅𝐵)
2		syl6breqr.2	. . 3 ⊢ 𝐶 = 𝐵
3	2	eqcomi 2092	. 2 ⊢ 𝐵 = 𝐶
4	1, 3	syl6breq 3882	1 ⊢ (𝜑 → 𝐴𝑅𝐶)

Syntax hints:   → wi 4   = wceq 1289   class class class wbr 3843

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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-un 3003  df-sn 3450  df-pr 3451  df-op 3453  df-br 3844

This theorem is referenced by:  fiunsnnn  6587  unsnfi  6619  exmidfodomrlemr  6818  exmidfodomrlemrALT  6819  gtndiv  8831  intqfrac2  9714  uzenom  9820  ege2le3  10948  eirraplem  11051