Theorem neleq12d 2468

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 2466	. . 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 2467	. . 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 188	1 |- ( ph -> ( A e/ C <-> B e/ D ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1364   e/ wnel 2462

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-5 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-4 1524  ax-17 1540  ax-ial 1548  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-cleq 2189  df-clel 2192  df-nel 2463

This theorem is referenced by: (None)