Theorem breqan12rd 4006

Description: Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.)

Hypotheses

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

Assertion

Ref	Expression
breqan12rd	|- ( ( ps /\ ph ) -> ( A R C <-> B R D ) )

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 4005	. 2 |- ( ( ph /\ ps ) -> ( A R C <-> B R D ) )
4	3	ancoms 266	1 |- ( ( ps /\ ph ) -> ( A R C <-> B R D ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1348   class class class wbr 3989

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990

This theorem is referenced by:  xltnegi  9792