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Theorem syl5breq 4841
Description: A chained equality inference for a binary relation. (Contributed by NM, 11-Oct-1999.)
Hypotheses
Ref Expression
syl5breq.1 𝐴𝑅𝐵
syl5breq.2 (𝜑𝐵 = 𝐶)
Assertion
Ref Expression
syl5breq (𝜑𝐴𝑅𝐶)

Proof of Theorem syl5breq
StepHypRef Expression
1 syl5breq.1 . . 3 𝐴𝑅𝐵
21a1i 11 . 2 (𝜑𝐴𝑅𝐵)
3 syl5breq.2 . 2 (𝜑𝐵 = 𝐶)
42, 3breqtrd 4830 1 (𝜑𝐴𝑅𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1632   class class class wbr 4804
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805
This theorem is referenced by:  syl5breqr  4842  phplem3  8308  xlemul1a  12331  phicl2  15695  sinq12ge0  24480  siilem1  28036  nmbdfnlbi  29238  nmcfnlbi  29241  unierri  29293  leoprf2  29316  leoprf  29317  ballotlemic  30898  ballotlem1c  30899  sumnnodd  40383
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