Theorem breqan12rd 2628

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 2627	. 2 |- ((ph /\ ps) -> (ARC <-> BRD))
4	3	ancoms 436	1 |- ((ps /\ ph) -> (ARC <-> BRD))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 954   class class class wbr 2614

This theorem is referenced by:  f1oweALT 3897  ltrpq 5065  ledivdivt 5846

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-v 1808  df-un 2046  df-sn 2408  df-pr 2409  df-op 2412  df-br 2615

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