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Theorem syl5eleq 2439
Description: B membership and equality inference. (Contributed by NM, 4-Jan-2006.)
Hypotheses
Ref Expression
syl5eleq.1 A B
syl5eleq.2 (φB = C)
Assertion
Ref Expression
syl5eleq (φA C)

Proof of Theorem syl5eleq
StepHypRef Expression
1 syl5eleq.1 . . 3 A B
21a1i 10 . 2 (φA B)
3 syl5eleq.2 . 2 (φB = C)
42, 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:  syl5eleqr  2440  enadj  6060
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