Theorem neleq1 2608

Description: Equality theorem for negated membership. (Contributed by NM, 20-Nov-1994.)

Assertion

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

Proof of Theorem neleq1

Step	Hyp	Ref	Expression
1		eleq1 2413	. . 3 |- (A = B -> (A e. C <-> B e. C))
2	1	notbid 285	. 2 |- (A = B -> (-. A e. C <-> -. B e. C))
3		df-nel 2520	. 2 |- (A e/ C <-> -. A e. C)
4		df-nel 2520	. 2 |- (B e/ C <-> -. B e. C)
5	2, 3, 4	3bitr4g 279	1 |- (A = B -> (A e/ C <-> B e/ C))

This proof depends on syntax axioms:  -. wn 3   -> wi 4   <-> wb 176   = wceq 1642   e. wcel 1710   e/ wnel 2518

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746  ax-ext 2334

This proof depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-cleq 2346  df-clel 2349  df-nel 2520

This theorem is used by:  neleq12d  2610