Theorem syl6breqr 3827

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 2086	. 2 ⊢ 𝐵 = 𝐶
4	1, 3	syl6breq 3826	1 ⊢ (𝜑 → 𝐴𝑅𝐶)

Syntax hints:   → wi 4   = wceq 1285   class class class wbr 3787

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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064

This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-un 2978  df-sn 3406  df-pr 3407  df-op 3409  df-br 3788

This theorem is referenced by:  fiunsnnn  6405  unsnfi  6429  gtndiv  8512  intqfrac2  9390  uzenom  9496