Theorem syl5breq 4690

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 4679	1 ⊢ (𝜑 → 𝐴𝑅𝐶)

Syntax hints:   → wi 4   = wceq 1483   class class class wbr 4653

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654

This theorem is referenced by:  syl5breqr  4691  phplem3  8141  xlemul1a  12118  phicl2  15473  sinq12ge0  24260  siilem1  27706  nmbdfnlbi  28908  nmcfnlbi  28911  unierri  28963  leoprf2  28986  leoprf  28987  ballotlemic  30568  ballotlem1c  30569  sumnnodd  39862