Theorem breqan12rd 3355

Description: Equality deduction for a binary relation.

Hypotheses

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

Assertion

Ref	Expression
breqan12rd	|- ((ps /\ ph) -> (ARC <-> BRD))

Proof of Theorem breqan12rd

Step	Hyp	Ref	Expression
1		breq1d.1	. . 3 |- (ph -> A = B)
2		breqan12i.2	. . 3 |- (ps -> C = D)
3	1, 2	breqan12d 3354	. 2 |- ((ph /\ ps) -> (ARC <-> BRD))
4	3	ancoms 484	1 |- ((ps /\ ph) -> (ARC <-> BRD))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   class class class wbr 3338

This theorem is referenced by:  f1oweALT 4883  ordtypelem7 5690  ltrpq 6237  ledivdiv 7073  ordtypelem7OLD 15381

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-v 2294  df-un 2600  df-sn 3049  df-pr 3050  df-op 3053  df-br 3339

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