Theorem syl6eqbrr 2653

Description: A chained equality inference for a binary relation.

Hypotheses

Ref	Expression
syl6eqbrr.1	|- (ph -> B = A)
syl6eqbrr.2	|- BRC

Assertion

Ref	Expression
syl6eqbrr	|- (ph -> ARC)

Proof of Theorem syl6eqbrr

Step	Hyp	Ref	Expression
1		syl6eqbrr.1	. . 3 |- (ph -> B = A)
2	1	eqcomd 1480	. 2 |- (ph -> A = B)
3		syl6eqbrr.2	. 2 |- BRC
4	2, 3	syl6eqbr 2652	1 |- (ph -> ARC)

Syntax hints:   -> wi 3   = wceq 956   class class class wbr 2619

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812  df-un 2050  df-sn 2412  df-pr 2413  df-op 2416  df-br 2620

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