Theorem neleq12d 2609

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 2607	. . 3 |- (A = B -> (A e/ C <-> B e/ C))
3	1, 2	syl 15	. 2 |- (ph -> (A e/ C <-> B e/ C))
4		neleq12d.2	. . 3 |- (ph -> C = D)
5		neleq2 2608	. . 3 |- (C = D -> (B e/ C <-> B e/ D))
6	4, 5	syl 15	. 2 |- (ph -> (B e/ C <-> B e/ D))
7	3, 6	bitrd 244	1 |- (ph -> (A e/ C <-> B e/ D))

Syntax hints:   -> wi 4   <-> wb 176   = wceq 1642   e/ wnel 2517

This theorem was proved from 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 theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-cleq 2346  df-clel 2349  df-nel 2519

This theorem is referenced by: (None)