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Theorem 3brtr4d 3821
Description: Substitution of equality into both sides of a binary relation. (Contributed by NM, 21-Feb-2005.)
Hypotheses
Ref Expression
3brtr4d.1 (𝜑𝐴𝑅𝐵)
3brtr4d.2 (𝜑𝐶 = 𝐴)
3brtr4d.3 (𝜑𝐷 = 𝐵)
Assertion
Ref Expression
3brtr4d (𝜑𝐶𝑅𝐷)

Proof of Theorem 3brtr4d
StepHypRef Expression
1 3brtr4d.1 . 2 (𝜑𝐴𝑅𝐵)
2 3brtr4d.2 . . 3 (𝜑𝐶 = 𝐴)
3 3brtr4d.3 . . 3 (𝜑𝐷 = 𝐵)
42, 3breq12d 3804 . 2 (𝜑 → (𝐶𝑅𝐷𝐴𝑅𝐵))
51, 4mpbird 160 1 (𝜑𝐶𝑅𝐷)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1259   class class class wbr 3791
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 2949  df-sn 3408  df-pr 3409  df-op 3411  df-br 3792
This theorem is referenced by:  f1oiso2  5493  prarloclemarch2  6574  caucvgprprlemmu  6850  caucvgsrlembound  6935  mulap0  7708  lediv12a  7934  recp1lt1  7939  fldiv4p1lem1div2  9249  intfracq  9264  modqmulnn  9286  addmodlteq  9342  frecfzennn  9361  monoord2  9394  expgt1  9452  leexp2r  9468  leexp1a  9469  bernneq  9530  faclbnd  9602  faclbnd6  9605  facubnd  9606  sqrtgt0  9853  absrele  9902  absimle  9903  abstri  9923  abs2difabs  9927  climsqz  10078  climsqz2  10079
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