Theorem breqtrd 2081

Description: Substitution of equal classes into a binary relation.

Hypotheses

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

Assertion

Ref	Expression
breqtrd	⊢ (φ → ARC)

Proof of Theorem breOtrd

Step	Hyp	Ref	Expression
1		breqtrd.1	. 2 ⊢ (φ → ARB)
2		breqtrd.2	. . 3 ⊢ (φ → B = C)
3	2	breq2d 2072	. 2 ⊢ (φ → (ARB ↔ ARC))
4	1, 3	mpbid 170	1 ⊢ (φ → ARC)

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

This theorem is referenced by:  breqtrrd 2083  phplem5 3407  discrlem3 4715  infmap2 4953  pjthlem10 5234  strlem3a 5693  cvexchlem 5759

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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