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Theorem syl6eleq 2130
Description: A membership and equality inference. (Contributed by NM, 4-Jan-2006.)
Hypotheses
Ref Expression
syl6eleq.1 (𝜑𝐴𝐵)
syl6eleq.2 𝐵 = 𝐶
Assertion
Ref Expression
syl6eleq (𝜑𝐴𝐶)

Proof of Theorem syl6eleq
StepHypRef Expression
1 syl6eleq.1 . 2 (𝜑𝐴𝐵)
2 syl6eleq.2 . . 3 𝐵 = 𝐶
32a1i 9 . 2 (𝜑𝐵 = 𝐶)
41, 3eleqtrd 2116 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:  syl6eleqr  2131  prid2g  3475  caucvgprprlem2  6806  gt0srpr  6831  eluzel2  8476  fseq1p1m1  8954  fznn0sub2  8983  nn0split  8992  exple1  9284  clim2iser  9830  clim2iser2  9831  iiserex  9832  iisermulc2  9833  iserile  9835  iserige0  9836  climub  9837  climserile  9838  serif0  9844  ialgrp1  9858
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