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Theorem 3brtr3d 3821
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 3805 . 2 (𝜑 → (𝐴𝑅𝐵𝐶𝑅𝐷))
51, 4mpbid 139 1 (𝜑𝐶𝑅𝐷)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1259   class class class wbr 3792
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-un 2950  df-sn 3409  df-pr 3410  df-op 3412  df-br 3793
This theorem is referenced by:  ofrval  5750  phplem2  6347  ltaddnq  6563  prarloclemarch2  6575  prmuloclemcalc  6721  axcaucvglemcau  7030  apreap  7652  ltmul1  7657  subap0d  7705  divap1d  7851  lemul2a  7900  monoord2  9400  expubnd  9477  bernneq2  9538  resqrexlemcalc2  9842  resqrexlemcalc3  9843  abs2dif2  9934
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