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Theorem syl5eleqr 2440
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 2358 . 2 (φB = C)
41, 3syl5eleq 2439 1 (φA C)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1642   wcel 1710
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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:  rabsnt  3797  enadj  6060  enprmaplem5  6080  ce0  6190  te0c  6237
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