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Theorem 3brtr4d 3794
 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 3777 . 2 (𝜑 → (𝐶𝑅𝐷𝐴𝑅𝐵))
51, 4mpbird 156 1 (𝜑𝐶𝑅𝐷)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1243   class class class wbr 3764 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765 This theorem is referenced by:  f1oiso2  5466  prarloclemarch2  6517  caucvgprprlemmu  6793  caucvgsrlembound  6878  mulap0  7635  lediv12a  7860  recp1lt1  7865  fldiv4p1lem1div2  9147  intfracq  9162  modqmulnn  9184  frecfzennn  9203  monoord2  9236  expgt1  9293  leexp2r  9308  leexp1a  9309  bernneq  9369  sqrtgt0  9632  absrele  9679  absimle  9680  abstri  9700  abs2difabs  9704  climsqz  9855  climsqz2  9856
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