Theorem neleq12d 2521

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 2519	. . 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 2520	. . 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 1402   e/ wnel 2515

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 623  ax-in2 624  ax-5 1500  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-4 1563  ax-17 1579  ax-ial 1587  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-cleq 2231  df-clel 2234  df-nel 2516

This theorem is referenced by: (None)