Theorem breqtrid 5147

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 5136	1 ⊢ (𝜑 → 𝐴𝑅𝐶)

This proof depends on syntax axioms:   → wi 4   = wceq 1569   class class class wbr 5108

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-rab 3416  df-v 3456  df-dif 3907  df-un 3909  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-br 5109

This theorem is used by:  breqtrrid  5148  xlemul1a  13320  phicl2  16833  sinq12ge0  26684  siilem1  31214  nmbdfnlbi  32412  nmcfnlbi  32415  unierri  32467  leoprf2  32490  leoprf  32491  2sqr3nconstr  34180  cos9thpinconstrlem2  34189  ballotlemic  34906  ballotlem1c  34907  sumnnodd  46374