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Theorem 3brtr4i 4030
Description: Substitution of equality into both sides of a binary relation. (Contributed by NM, 11-Aug-1999.)
Hypotheses
Ref Expression
3brtr4.1 𝐴𝑅𝐵
3brtr4.2 𝐶 = 𝐴
3brtr4.3 𝐷 = 𝐵
Assertion
Ref Expression
3brtr4i 𝐶𝑅𝐷

Proof of Theorem 3brtr4i
StepHypRef Expression
1 3brtr4.2 . . 3 𝐶 = 𝐴
2 3brtr4.1 . . 3 𝐴𝑅𝐵
31, 2eqbrtri 4021 . 2 𝐶𝑅𝐵
4 3brtr4.3 . 2 𝐷 = 𝐵
53, 4breqtrri 4027 1 𝐶𝑅𝐷
Colors of variables: wff set class
Syntax hints:   = wceq 1353   class class class wbr 4000
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-un 3133  df-sn 3597  df-pr 3598  df-op 3600  df-br 4001
This theorem is referenced by:  1lt2nq  7393  0lt1sr  7752  ax0lt1  7863  declt  9397  decltc  9398  decle  9403  frecfzennn  10409  fsumabs  11454  2strbasg  12554  2stropg  12555  basendxlttsetndx  12609  basendxltdsndx  12626
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