Theorem breqan12d 2632

Description: Equality deduction for a binary relation.

Hypotheses

Ref	Expression
breq1d.1	|- (ph -> A = B)
breqan12i.2	|- (ps -> C = D)

Assertion

Ref	Expression
breqan12d	|- ((ph /\ ps) -> (ARC <-> BRD))

Proof of Theorem breqan12d

Step	Hyp	Ref	Expression
1		breq12 2624	. 2 |- ((A = B /\ C = D) -> (ARC <-> BRD))
2		breq1d.1	. 2 |- (ph -> A = B)
3		breqan12i.2	. 2 |- (ps -> C = D)
4	1, 2, 3	syl2an 454	1 |- ((ph /\ ps) -> (ARC <-> BRD))

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

This theorem is referenced by:  breqan12rd 2633  isoid 3895  isotr 3897  isotrALT 3898  oprec 4318  pre-axltadd 5289  leltaddt 5646  lemul1it 5837  lemul1itOLD 5838  expwordit 6603  lt2sqt 6630  le2sqt 6631  sqrle 6707  sqrlt 6708  ser1f0 7170  minveclem26 8570  minveclem27 8571  logltbt 8776  projlemHIL 9218  mddmdt 10228

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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