Theorem 3brtr4d 3815

Description: Substitution of equality into both sides of a binary relation. (Contributed by NM, 21-Feb-2005.)

Hypotheses

Ref	Expression
3brtr4d.1	|- ( ph -> A R B )
3brtr4d.2	|- ( ph -> C = A )
3brtr4d.3	|- ( ph -> D = B )

Assertion

Ref	Expression
3brtr4d	|- ( ph -> C R D )

Proof of Theorem 3brtr4d

Step	Hyp	Ref	Expression
1		3brtr4d.1	. 2 |- ( ph -> A R B )
2		3brtr4d.2	. . 3 |- ( ph -> C = A )
3		3brtr4d.3	. . 3 |- ( ph -> D = B )
4	2, 3	breq12d 3798	. 2 |- ( ph -> ( C R D <-> A R B ) )
5	1, 4	mpbird 165	1 |- ( ph -> C R D )

Syntax hints:   -> wi 4   = wceq 1284   class class class wbr 3785

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-un 2977  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786

This theorem is referenced by:  f1oiso2  5486  prarloclemarch2  6609  caucvgprprlemmu  6885  caucvgsrlembound  6970  mulap0  7744  lediv12a  7972  recp1lt1  7977  fldiv4p1lem1div2  9307  intfracq  9322  modqmulnn  9344  addmodlteq  9400  frecfzennn  9419  monoord2  9456  expgt1  9514  leexp2r  9530  leexp1a  9531  bernneq  9593  faclbnd  9668  faclbnd6  9671  facubnd  9672  sqrtgt0  9920  absrele  9969  absimle  9970  abstri  9990  abs2difabs  9994  climsqz  10173  climsqz2  10174