Theorem syl6breq 2654

Description: A chained equality inference for a binary relation.

Hypotheses

Ref	Expression
syl6breq.1	|- (ph -> ARB)
syl6breq.2	|- B = C

Assertion

Ref	Expression
syl6breq	|- (ph -> ARC)

Proof of Theorem syl6breq

Step	Hyp	Ref	Expression
1		syl6breq.1	. 2 |- (ph -> ARB)
2		eqid 1475	. 2 |- A = A
3		syl6breq.2	. 2 |- B = C
4	1, 2, 3	3brtr3g 2646	1 |- (ph -> ARC)

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

This theorem is referenced by:  syl6breqr 2655  ltbtwnpq 5084  1pr 5117  prlem934 5139  ltexprlem2 5143  msqgt0 5613  recgt0i 5814  zltp1let 6181  exple1t 6607  abs3lem 6901  faclbnd4lem1 6948  isumclim3t 7200  ivthlem1 7281  ivthlem6 7286  efcvg 7314  cos01gt0 7477  sin02gt0 7478  infcda 7567  infxp 7572  alephadd 7582  minveclem30 8574  sineq0 8713  norm3lem 9016  projlem12 9197  nmopadjlem 10022  nmopcoadj 10034  hstlet 10157  stadd3 10175  strlem3a 10179  strlem5 10182

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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