Theorem breqtr 2080

Description: Substitution of equal classes into a binary relation.

Hypotheses

Ref	Expression
breqtr.1	⊢ ARB
breqtr.2	⊢ B = C

Assertion

Ref	Expression
breqtr	⊢ ARC

Proof of Theorem breqtr

Step	Hyp	Ref	Expression
1		breqtr.1	. 2 ⊢ ARB
2		breqtr.2	Ü . 3 ⊢ B = C
3	2	breq2i 2069	. 2 ⊢ (ARB ↔ ARC)
4	1, 3	mpbi 164	1 ⊢ ARC

Syntax hints:   = wceq 1091   class class class wbr 2054

This theorem is referenced by:  breqtrr 2082  3brtr3 2084  cdacomen 3724  cdaassen 3725  lt01 4377  nn0addge2 4561  sqrlem10 4740  sqrlem11 4741  abslt 4855  absle 4856  abstri 4859  ruclem30 4914  normlem5 5067  normlem6 5068  norm-ii 5086  norm3adif 5095  projlem3 5195  projlem18 5210  cmm2 5515

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-un 1490  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063

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