Theorem syl6eqbrr 3802

Description: A chained equality inference for a binary relation. (Contributed by NM, 4-Jan-2006.)

Hypotheses

Ref	Expression
syl6eqbrr.1	⊢ (𝜑 → 𝐵 = 𝐴)
syl6eqbrr.2	⊢ 𝐵𝑅𝐶

Assertion

Ref	Expression
syl6eqbrr	⊢ (𝜑 → 𝐴𝑅𝐶)

Proof of Theorem syl6eqbrr

Step	Hyp	Ref	Expression
1		syl6eqbrr.1	. . 3 ⊢ (𝜑 → 𝐵 = 𝐴)
2	1	eqcomd 2045	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
3		syl6eqbrr.2	. 2 ⊢ 𝐵𝑅𝐶
4	2, 3	syl6eqbr 3801	1 ⊢ (𝜑 → 𝐴𝑅𝐶)

Syntax hints:   → wi 4   = wceq 1243   class class class wbr 3764

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765

This theorem is referenced by: (None)