Theorem breqtrid 5107

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

Hypotheses

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

Assertion

Ref	Expression
breqtrid	⊢ (𝜑 → 𝐴𝑅𝐶)

Proof of Theorem breqtrid

Step	Hyp	Ref	Expression
1		breqtrid.1	. . 3 ⊢ 𝐴𝑅𝐵
2	1	a1i 11	. 2 ⊢ (𝜑 → 𝐴𝑅𝐵)
3		breqtrid.2	. 2 ⊢ (𝜑 → 𝐵 = 𝐶)
4	2, 3	breqtrd 5096	1 ⊢ (𝜑 → 𝐴𝑅𝐶)

Syntax hints:   → wi 4   = wceq 1539   class class class wbr 5070

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071

This theorem is referenced by:  breqtrrid  5108  phplem3  8894  xlemul1a  12951  phicl2  16397  sinq12ge0  25570  siilem1  29114  nmbdfnlbi  30312  nmcfnlbi  30315  unierri  30367  leoprf2  30390  leoprf  30391  ballotlemic  32373  ballotlem1c  32374  sumnnodd  43061