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Theorem 3brtr3g 4487
Description: Substitution of equality into both sides of a binary relation. (Contributed by NM, 16-Jan-1997.)
Hypotheses
Ref Expression
3brtr3g.1  |-  ( ph  ->  A R B )
3brtr3g.2  |-  A  =  C
3brtr3g.3  |-  B  =  D
Assertion
Ref Expression
3brtr3g  |-  ( ph  ->  C R D )

Proof of Theorem 3brtr3g
StepHypRef Expression
1 3brtr3g.1 . 2  |-  ( ph  ->  A R B )
2 3brtr3g.2 . . 3  |-  A  =  C
3 3brtr3g.3 . . 3  |-  B  =  D
42, 3breq12i 4465 . 2  |-  ( A R B  <->  C R D )
51, 4sylib 196 1  |-  ( ph  ->  C R D )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1395   class class class wbr 4456
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-br 4457
This theorem is referenced by:  syl5eqbrr  4490  syl6breq  4495  ssenen  7710  adderpq  9351  mulerpq  9352  ltaddnq  9369  ege2le3  13837  ovolfiniun  22038  dvfsumlem3  22555  basellem9  23488  pnt2  23924  pnt  23925  siilem1  25893  omndaddr  27857  ogrpaddltrd  27870
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