Theorem eqbrtrd 2077

Description: Substitution of equal classes into a binary relation.

Hypotheses

Ref	Expression
eqbrtrd.1	|- (ph -> A = B)
eqbrtrd.2	|- (ph -> BRC)

Assertion

Ref	Expression
eqbrtrd	|- (ph -> ARC)

Proof of Theorem eqbrtrd

Step	Hyp	Ref	Expression
1		eqbrtrd.2	. 2 |- (ph -> BRC)
2		eqbrtrd.1	. . 3 |- (ph -> A = B)
3	2	breq1d 2071	. 2 |- (ph -> (ARC <-> BRC))
4	1, 3	mpbird 171	1 |- (ph -> ARC)

Syntax hints:   -> wi 2   = wceq 1091   class class class wbr 2054

This theorem is referenced by:  eqbrtrrd 2079  cdafi 3730  ruclem27 4911  alephsuc3 4955  bcs 5101  projlem26 5218  pjs14 5669  strlem3a 5693

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

metamath.org