Theorem neleq2 1643

Description: Equality theorem for negated membership.

Assertion

Ref	Expression
neleq2	|- (A = B -> (C e/ A <-> C e/ B))

Proof of Theorem neleq2

Step	Hyp	Ref	Expression
1		eleq2 1535	. . 3 |- (A = B -> (C e. A <-> C e. B))
2	1	negbid 611	. 2 |- (A = B -> (-. C e. A <-> -. C e. B))
3		df-nel 1588	. 2 |- (C e/ A <-> -. C e. A)
4		df-nel 1588	. 2 |- (C e/ B <-> -. C e. B)
5	2, 3, 4	3bitr4g 555	1 |- (A = B -> (C e/ A <-> C e/ B))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   = wceq 956   e. wcel 958   e/ wnel 1586

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 963  ax-17 971  ax-4 973  ax-5o 975  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-cleq 1469  df-clel 1472  df-nel 1588

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