Theorem neleq12d 2901

Description: Equality theorem for negated membership. (Contributed by FL, 10-Aug-2016.) (Proof shortened by Wolf Lammen, 25-Nov-2019.)

Hypotheses

Ref	Expression
neleq12d.1	|- ( ph -> A = B )
neleq12d.2	|- ( ph -> C = D )

Assertion

Ref	Expression
neleq12d	|- ( ph -> ( A e/ C <-> B e/ D ) )

Proof of Theorem neleq12d

Step	Hyp	Ref	Expression
1		neleq12d.1	. . . 4 |- ( ph -> A = B )
2		neleq12d.2	. . . 4 |- ( ph -> C = D )
3	1, 2	eleq12d 2695	. . 3 |- ( ph -> ( A e. C <-> B e. D ) )
4	3	notbid 308	. 2 |- ( ph -> ( -. A e. C <-> -. B e. D ) )
5		df-nel 2898	. 2 |- ( A e/ C <-> -. A e. C )
6		df-nel 2898	. 2 |- ( B e/ D <-> -. B e. D )
7	4, 5, 6	3bitr4g 303	1 |- ( ph -> ( A e/ C <-> B e/ D ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   = wceq 1483   e. wcel 1990   e/ wnel 2897

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-cleq 2615  df-clel 2618  df-nel 2898

This theorem is referenced by:  neleq1  2902  neleq2  2903  uhgrspan1  26195  nbgrnself  26257  nbgrnself2  26259  finsumvtxdg2size  26446