Theorem neleq1 1639

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 1531	. . 3 |- (A = B -> (A e. C <-> B e. C))
2	1	negbid 610	. 2 |- (A = B -> (-. A e. C <-> -. B e. C))
3		df-nel 1585	. 2 |- (A e/ C <-> -. A e. C)
4		df-nel 1585	. 2 |- (B e/ C <-> -. B e. C)
5	2, 3, 4	3bitr4g 554	1 |- (A = B -> (A e/ C <-> B e/ C))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   = wceq 954   e. wcel 956   e/ wnel 1583

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 961  ax-17 969  ax-4 971  ax-5o 973  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 979  df-cleq 1467  df-clel 1470  df-nel 1585

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