Theorem neleq1 1199

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 1149	. . 3 |- (A = B -> (A e. C <-> B e. C))
2	1	negbid 463	. 2 |- (A = B -> (-. A e. C <-> -. B e. C))
3		df-nel 1193	. 2 |- (A e/ C <-> -. A e. C)
4		df-nel 1193	. 2 |- (B e/ C <-> -. B e. C)
5	2, 3, 4	3bitr4g 428	1 |- (A = B -> (A e/ C <-> B e/ C))

Syntax hints:  -. wn 1   -> wi 2   <-> wb 127   = wceq 1091   e. wcel 1092   e/ wnel 1191

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-gen 677  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-cleq 1097  df-clel 1099  df-nel 1193

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