Theorem breqan12rd 3978

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 3977	. 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 1332   class class class wbr 3961

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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-ext 2136

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1740  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-v 2711  df-un 3102  df-sn 3562  df-pr 3563  df-op 3565  df-br 3962

This theorem is referenced by:  xltnegi  9717