Theorem breqtrid 5141

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 5130	1 ⊢ (𝜑 → 𝐴𝑅𝐶)

This proof depends on syntax axioms:   → wi 4   = wceq 1569   class class class wbr 5102

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-br 5103

This theorem is used by:  breqtrrid  5142  xlemul1a  13418  phicl2  16945  sinq12ge0  26837  siilem1  31453  nmbdfnlbi  32651  nmcfnlbi  32654  unierri  32706  leoprf2  32729  leoprf  32730  2sqr3nconstr  34413  cos9thpinconstrlem2  34422  ballotlemic  35139  ballotlem1c  35140  sumnnodd  46642