Theorem neleq12d 2345

Description: Equality theorem for negated membership. (Contributed by FL, 10-Aug-2016.)

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	. . 3 |- ( ph -> A = B )
2		neleq1 2343	. . 3 |- ( A = B -> ( A e/ C <-> B e/ C ) )
3	1, 2	syl 14	. 2 |- ( ph -> ( A e/ C <-> B e/ C ) )
4		neleq12d.2	. . 3 |- ( ph -> C = D )
5		neleq2 2344	. . 3 |- ( C = D -> ( B e/ C <-> B e/ D ) )
6	4, 5	syl 14	. 2 |- ( ph -> ( B e/ C <-> B e/ D ) )
7	3, 6	bitrd 186	1 |- ( ph -> ( A e/ C <-> B e/ D ) )

Syntax hints:   -> wi 4   <-> wb 103   = wceq 1284   e/ wnel 2339

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-5 1376  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-4 1440  ax-17 1459  ax-ial 1467  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-cleq 2074  df-clel 2077  df-nel 2340

This theorem is referenced by: (None)