Theorem breqtrid 5142

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 5131	1 ⊢ (𝜑 → 𝐴𝑅𝐶)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   class class class wbr 5103

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104

This theorem is used by:  breqtrrid  5143  xlemul1a  13373  phicl2  16892  sinq12ge0  26786  siilem1  31372  nmbdfnlbi  32570  nmcfnlbi  32573  unierri  32625  leoprf2  32648  leoprf  32649  2sqr3nconstr  34332  cos9thpinconstrlem2  34341  ballotlemic  35059  ballotlem1c  35060  sumnnodd  46558