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Theorem syl5eleqr 2124
 Description: B membership and equality inference. (Contributed by NM, 4-Jan-2006.)
Hypotheses
Ref Expression
syl5eleqr.1 A B
syl5eleqr.2 (φ𝐶 = B)
Assertion
Ref Expression
syl5eleqr (φA 𝐶)

Proof of Theorem syl5eleqr
StepHypRef Expression
1 syl5eleqr.1 . 2 A B
2 syl5eleqr.2 . . 3 (φ𝐶 = B)
32eqcomd 2042 . 2 (φB = 𝐶)
41, 3syl5eleq 2123 1 (φA 𝐶)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1242   ∈ wcel 1390 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-5 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-4 1397  ax-17 1416  ax-ial 1424  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-cleq 2030  df-clel 2033 This theorem is referenced by:  rabsnt  3436  0elnn  4283  tfrexlem  5889  rdgtfr  5901  rdgruledefgg  5902
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