Theorem breqtrrd 2083

Description: Substitution of equal classes into a binary relation.

Hypotheses

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

Assertion

Ref	Expression
breqtrrd	⊢ (φ → ARC)

Proof of Theorem breqtrrd

Step	Hyp	Ref	Expression
1		breqtrrd.1	. 2 ⊢ (φ → ARB)
2		breqtrrd.2	. . 3 ⊢ (φ → C = B)
3	2	cleqcomd 1106	. 2 ⊢ (φ → B = C)
4	1, 3	breqtrd 2081	1 ⊢ (φ → ARC)

Syntax hints:   → wi 2   = wceq 1091   class class class wbr 2054

This theorem is referenced by:  ruclem26 4910  ruclem28 4912  normge0t 5077  normpyct 5093  stle 5681  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

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