Theorem breq12 2614

Description: Equality theorem for a binary relation.

Assertion

Ref	Expression
breq12	|- ((A = B /\ C = D) -> (ARC <-> BRD))

Proof of Theorem breq12

Step	Hyp	Ref	Expression
1		breq1 2612	. 2 |- (A = B -> (ARC <-> BRC))
2		breq2 2613	. 2 |- (C = D -> (BRC <-> BRD))
3	1, 2	sylan9bb 538	1 |- ((A = B /\ C = D) -> (ARC <-> BRD))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 953   class class class wbr 2609

This theorem is referenced by:  breq12i 2618  breqan12d 2622  ersym 4256  canth2g 4466  zorn2lem6 4765  brdom6disj 4777  ltresr 5230  xrltnrt 5514  xrltnsymt 5523  xrlttrit 5525  xrlttrt 5526  qbtwnxr 6217  pslem 8573

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-v 1803  df-un 2040  df-sn 2402  df-pr 2403  df-op 2406  df-br 2610

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