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Theorem 3brtr3d 3756
Description: Substitution of equality into both sides of a binary relation. (Contributed by NM, 18-Oct-1999.)
Hypotheses
Ref Expression
3brtr3d.1 (φA𝑅B)
3brtr3d.2 (φA = 𝐶)
3brtr3d.3 (φB = 𝐷)
Assertion
Ref Expression
3brtr3d (φ𝐶𝑅𝐷)

Proof of Theorem 3brtr3d
StepHypRef Expression
1 3brtr3d.1 . 2 (φA𝑅B)
2 3brtr3d.2 . . 3 (φA = 𝐶)
3 3brtr3d.3 . . 3 (φB = 𝐷)
42, 3breq12d 3740 . 2 (φ → (A𝑅B𝐶𝑅𝐷))
51, 4mpbid 135 1 (φ𝐶𝑅𝐷)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1223   class class class wbr 3727
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 614  ax-5 1309  ax-7 1310  ax-gen 1311  ax-ie1 1355  ax-ie2 1356  ax-8 1368  ax-10 1369  ax-11 1370  ax-i12 1371  ax-bnd 1372  ax-4 1373  ax-17 1392  ax-i9 1396  ax-ial 1400  ax-i5r 1401  ax-ext 1995
This theorem depends on definitions:  df-bi 110  df-3an 869  df-tru 1226  df-nf 1323  df-sb 1619  df-clab 2000  df-cleq 2006  df-clel 2009  df-nfc 2140  df-v 2528  df-un 2890  df-sn 3345  df-pr 3346  df-op 3348  df-br 3728
This theorem is referenced by:  ofrval  5633  ltaddnq  6251  prarloclemarch2  6262  prmuloclemcalc  6395  apreap  7174  ltmul1  7179  divap1d  7361  lemul2a  7408
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