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Theorem syl6eqbrr 4725
Description: A chained equality inference for a binary relation. (Contributed by NM, 4-Jan-2006.)
Hypotheses
Ref Expression
syl6eqbrr.1 (𝜑𝐵 = 𝐴)
syl6eqbrr.2 𝐵𝑅𝐶
Assertion
Ref Expression
syl6eqbrr (𝜑𝐴𝑅𝐶)

Proof of Theorem syl6eqbrr
StepHypRef Expression
1 syl6eqbrr.1 . . 3 (𝜑𝐵 = 𝐴)
21eqcomd 2657 . 2 (𝜑𝐴 = 𝐵)
3 syl6eqbrr.2 . 2 𝐵𝑅𝐶
42, 3syl6eqbr 4724 1 (𝜑𝐴𝑅𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1523   class class class wbr 4685
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686
This theorem is referenced by:  grur1  9680  t1connperf  21287  basellem9  24860  sqff1o  24953  ballotlemic  30696  ballotlem1c  30697
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