Theorem syl6breq 2093

Description: A chained equality inference for a binary relation.

Hypotheses

Ref	Expression
syl6breq.1	⊢ (φ → ARB)
syl6breq.2	⊢ B = C

Assertion

Ref	Expression
syl6breq	⊢ (φ → ARC)

Proof of Theorem syl6breq

Step	Hyp	Ref	Expression
1		syl6breq.1	. 2 ⊢ (φ → ARB)
2		cleqid 1102	. 2 ⊢ A = A
3		syl6breq.2	. 2 ⊢ B = C
4	1, 2, 3	3brtr3g 2087	1 ⊢ (φ → ARC)

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

This theorem is referenced by:  ltbtwnpq 3878  1pr 3911  prlem934 3933  ltexprlem2 3937  sqgt0 4343  recgt0i 4385  zltp1let 4597  abs3lem 4861  infunabs 4946  infcdaabs 4947  norm3lem 5096  projlem12 5204  stadd 5687  stadd3 5689  strlem3a 5693  strlem5 5696

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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