Theorem syl6eqbr 2092

Description: A chained equality inference for a binary relation.

Hypotheses

Ref	Expression
syl6eqbr.1	⊢ (φ → A = B)
syl6eqbr.2	⊢ BRC

Assertion

Ref	Expression
syl6eqbr	⊢ (φ → ARC)

Proof of Theorem syl6eqbr

Step	Hyp	Ref	Expression
1		syl6eqbr.2	. 2 ⊢ BRC
2		syl6eqbr.1	. . 3 ⊢ (φ → A = B)
3	2	breq1d 2071	. 2 ⊢ (φ → (ARC ↔ BRC))
4	1, 3	mpbiri 169	1 ⊢ (φ → ARC)

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

This theorem is referenced by:  mapdom2 3389  ruclem27 4911  bcs 5101  occllem5 5184

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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