Theorem eqbrtr 2076

Description: Substitution of equal classes into a binary relation.

Hypotheses

Ref	Expression
eqbrtr.1	⊢ A = B
eqbrtr.2	⊢ BRC

Assertion

Ref	Expression
eqbrtr	⊢ ARC

Proof of Theorem eqbrtr

Step	Hyp	Ref	Expression
1		eqbrtr.2	. 2 ⊢ BRC
2		eqbrtr.1	. . 3 ⊢ A = B
3	2	breq1i 2068	. 2 ⊢ (ARC ↔ BRC)
4	1, 3	mpbir 165	1 ⊢ ARC

Syntax hints:   = wceq 1091   class class class wbr 2054

This theorem is referenced by:  eqbrtrr 2078  3brtr4 2085  aleph1 3676  cda0en 3720  xp1en 3722  halfnz 4586  sqrlem6 4736  sqrlem10 4740  sqrlem11 4741  sqrlem19 4749  nthruz 4785  abs3dif 4860  ruclem31 4915  ruclem32 4916  norm3dif 5094  norm3adif 5095  bcs 5101  occllem1 5180  projlem3 5195  projlem5 5197  projlem7 5199  projlem18 5210

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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