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Theorem 3brtr4d 3785
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 3768 . 2 (φ → (𝐶𝑅𝐷A𝑅B))
51, 4mpbird 156 1 (φ𝐶𝑅𝐷)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1242   class class class wbr 3755
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756
This theorem is referenced by:  f1oiso2  5409  prarloclemarch2  6402  mulap0  7377  lediv12a  7601  recp1lt1  7606  frecfzennn  8844  expgt1  8907  leexp2r  8922  leexp1a  8923  bernneq  8982
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