Theorem neleq1 1634

Description: Equality theorem for negated membership.

Assertion

Ref	Expression
neleq1	|- (A = B -> (A e/ C <-> B e/ C))

Proof of Theorem neleq1

Step	Hyp	Ref	Expression
1		eleq1 1526	. . 3 |- (A = B -> (A e. C <-> B e. C))
2	1	negbid 609	. 2 |- (A = B -> (-. A e. C <-> -. B e. C))
3		df-nel 1580	. 2 |- (A e/ C <-> -. A e. C)
4		df-nel 1580	. 2 |- (B e/ C <-> -. B e. C)
5	2, 3, 4	3bitr4g 553	1 |- (A = B -> (A e/ C <-> B e/ C))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   = wceq 953   e. wcel 955   e/ wnel 1578

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 960  ax-17 968  ax-4 970  ax-5o 972  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 978  df-cleq 1462  df-clel 1465  df-nel 1580

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