Theorem syl6breq 2659

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 1478	. 2 |- A = A
3		syl6breq.2	. 2 |- B = C
4	1, 2, 3	3brtr3g 2651	1 |- (ph -> ARC)

Syntax hints:   -> wi 3   = wceq 958   class class class wbr 2624

This theorem is referenced by:  syl6breqr 2660  ltbtwnpq 5096  1pr 5129  prlem934 5151  ltexprlem2 5155  msqgt0 5625  recgt0i 5816  zltp1let 6183  exple1t 6608  abs3lem 6901  faclbnd4lem1 6948  isumclim3t 7200  ivthlem1 7281  ivthlem6 7286  efcvg 7314  cos01gt0 7478  sin02gt0 7479  infcda 7568  infxp 7573  alephadd 7584  minveclem30 8570  sineq0 8708  norm3lem 9011  projlem12 9192  nmopadjlem 10017  nmopcoadj 10029  hstlet 10152  stadd3 10170  strlem3a 10174  strlem5 10177

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-v 1815  df-un 2053  df-sn 2416  df-pr 2417  df-op 2420  df-br 2625

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