Theorem breq12 2624

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 2622	. 2 |- (A = B -> (ARC <-> BRC))
2		breq2 2623	. 2 |- (C = D -> (BRC <-> BRD))
3	1, 2	sylan9bb 540	1 |- ((A = B /\ C = D) -> (ARC <-> BRD))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 956   class class class wbr 2619

This theorem is referenced by:  breq12i 2628  breqan12d 2632  ersym 4272  canth2g 4485  zorn2lem6 4793  brdom6disj 4805  ltresr 5258  xrltnrt 5541  xrltnsymt 5550  xrlttrit 5552  xrlttrt 5553  qbtwnxr 6279  pslem 8647

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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