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Theorem syl5eleqr 2127
 Description: B membership and equality inference. (Contributed by NM, 4-Jan-2006.)
Hypotheses
Ref Expression
syl5eleqr.1 𝐴𝐵
syl5eleqr.2 (𝜑𝐶 = 𝐵)
Assertion
Ref Expression
syl5eleqr (𝜑𝐴𝐶)

Proof of Theorem syl5eleqr
StepHypRef Expression
1 syl5eleqr.1 . 2 𝐴𝐵
2 syl5eleqr.2 . . 3 (𝜑𝐶 = 𝐵)
32eqcomd 2045 . 2 (𝜑𝐵 = 𝐶)
41, 3syl5eleq 2126 1 (𝜑𝐴𝐶)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1243   ∈ wcel 1393 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 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-17 1419  ax-ial 1427  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-cleq 2033  df-clel 2036 This theorem is referenced by:  rabsnt  3445  0elnn  4340  tfrexlem  5948  rdgtfr  5961  rdgruledefgg  5962
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