Theorem syl5breqr 2651

Description: A chained equality inference for a binary relation.

Hypotheses

Ref	Expression
syl5breqr.1	|- (ph -> B = A)
syl5breqr.2	|- CRA

Assertion

Ref	Expression
syl5breqr	|- (ph -> CRB)

Proof of Theorem syl5breqr

Step	Hyp	Ref	Expression
1		syl5breqr.1	. . 3 |- (ph -> B = A)
2	1	eqcomd 1480	. 2 |- (ph -> A = B)
3		syl5breqr.2	. 2 |- CRA
4	2, 3	syl5breq 2650	1 |- (ph -> CRB)

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

This theorem is referenced by:  alephordlem1 4872  expgt0t 6589  expge0t 6591  expge1t 6593  sqlecant 6641  bernneq 6652  cvgcmp3cetlem1 7188  cvgcmp3cetlem2 7189  eirrlem4 7392  efgt0 7404  ruclem26 7535  dscmet 7918  pjssge0 9627  unierr 10037

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

Copyright terms: Public domain