Theorem neleq2 2102

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 1958	. . 3 |- (A = B -> (C e. A <-> C e. B))
2	1	notbid 673	. 2 |- (A = B -> (-. C e. A <-> -. C e. B))
3		df-nel 2020	. 2 |- (C e/ A <-> -. C e. A)
4		df-nel 2020	. 2 |- (C e/ B <-> -. C e. B)
5	2, 3, 4	3bitr4g 614	1 |- (A = B -> (C e/ A <-> C e/ B))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   = wceq 1298   e. wcel 1300   e/ wnel 2018

This theorem is referenced by:  isfbas 10261  bnj25 12392  regsep 15550  fcluscomp 15621

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 1305  ax-17 1317  ax-4 1319  ax-5o 1321  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-an 242  df-ex 1327  df-cleq 1877  df-clel 1880  df-nel 2020

Copyright terms: Public domain