Theorem breqan12d 2074

Description: Equality deduction for 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 2067	. 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 349	1 |- ((ph /\ ps) -> (ARC <-> BRD))

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196   = wceq 1091   class class class wbr 2054

This theorem is referenced by:  breqan12rd 2075  isoid 2933  oprec 3254  axltadd 4085  le2sqet 4707  sqrle 4765

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-un 1490  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063

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