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Theorem syl5breqr 3791
Description: B chained equality inference for a binary relation. (Contributed by NM, 24-Apr-2005.)
Hypotheses
Ref Expression
syl5breqr.1 A𝑅B
syl5breqr.2 (φ𝐶 = B)
Assertion
Ref Expression
syl5breqr (φA𝑅𝐶)

Proof of Theorem syl5breqr
StepHypRef Expression
1 syl5breqr.1 . 2 A𝑅B
2 syl5breqr.2 . . 3 (φ𝐶 = B)
32eqcomd 2042 . 2 (φB = 𝐶)
41, 3syl5breq 3790 1 (φA𝑅𝐶)
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-bndl 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:  bernneq  9022
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