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Theorem syl6eleq 2443
 Description: A membership and equality inference. (Contributed by NM, 4-Jan-2006.)
Hypotheses
Ref Expression
syl6eleq.1 (φA B)
syl6eleq.2 B = C
Assertion
Ref Expression
syl6eleq (φA C)

Proof of Theorem syl6eleq
StepHypRef Expression
1 syl6eleq.1 . 2 (φA B)
2 syl6eleq.2 . . 3 B = C
32a1i 10 . 2 (φB = C)
41, 3eleqtrd 2429 1 (φA C)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1642   ∈ wcel 1710 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-cleq 2346  df-clel 2349 This theorem is referenced by:  syl6eleqr  2444  prid2g  3826  sfinltfin  4535  vinf  4555  ndmfvrcl  5350  enmap2lem3  6065  enmap1lem3  6071  enprmaplem3  6078  enprmaplem5  6080  enprmaplem6  6081  eqncg  6126  ncseqnc  6128
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