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Theorem syl5breq 4614
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 4603 1 (𝜑𝐴𝑅𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1474   class class class wbr 4577
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-rab 2904  df-v 3174  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-br 4578
This theorem is referenced by:  syl5breqr  4615  phplem3  8003  xlemul1a  11947  phicl2  15257  sinq12ge0  23981  siilem1  26896  nmbdfnlbi  28098  nmcfnlbi  28101  unierri  28153  leoprf2  28176  leoprf  28177  ballotlemic  29701  ballotlem1c  29702  sumnnodd  38494
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