Theorem breqtrrid 4027

Description: B chained equality inference for a binary relation. (Contributed by NM, 24-Apr-2005.)

Hypotheses

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

Assertion

Ref	Expression
breqtrrid	⊢ (𝜑 → 𝐴𝑅𝐶)

Proof of Theorem breqtrrid

Step	Hyp	Ref	Expression
1		breqtrrid.1	. 2 ⊢ 𝐴𝑅𝐵
2		breqtrrid.2	. . 3 ⊢ (𝜑 → 𝐶 = 𝐵)
3	2	eqcomd 2176	. 2 ⊢ (𝜑 → 𝐵 = 𝐶)
4	1, 3	breqtrid 4026	1 ⊢ (𝜑 → 𝐴𝑅𝐶)

Syntax hints:   → wi 4   = wceq 1348   class class class wbr 3989

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990

This theorem is referenced by:  xsubge0  9838  xposdif  9839  bernneq  10596  pcge0  12266  rpabscxpbnd  13653  lgsdir2lem2  13724  trilpolemclim  14068  trilpolemlt1  14073  nconstwlpolemgt0  14095