Theorem syl5breq 4620

Description: A chained equality inference for a binary relation. (Contributed by NM, 11-Oct-1999.)

Hypotheses

Ref	Expression
syl5breq.1	⊢ 𝐴𝑅𝐵
syl5breq.2	⊢ (𝜑 → 𝐵 = 𝐶)

Assertion

Ref	Expression
syl5breq	⊢ (𝜑 → 𝐴𝑅𝐶)

Proof of Theorem syl5breq

Step	Hyp	Ref	Expression
1		syl5breq.1	. . 3 ⊢ 𝐴𝑅𝐵
2	1	a1i 11	. 2 ⊢ (𝜑 → 𝐴𝑅𝐵)
3		syl5breq.2	. 2 ⊢ (𝜑 → 𝐵 = 𝐶)
4	2, 3	breqtrd 4609	1 ⊢ (𝜑 → 𝐴𝑅𝐶)

Syntax hints:   → wi 4   = wceq 1475   class class class wbr 4583

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584

This theorem is referenced by:  syl5breqr  4621  phplem3  8026  xlemul1a  11990  phicl2  15311  sinq12ge0  24064  siilem1  27090  nmbdfnlbi  28292  nmcfnlbi  28295  unierri  28347  leoprf2  28370  leoprf  28371  ballotlemic  29895  ballotlem1c  29896  sumnnodd  38697