Theorem necon3bid 2246

Description: Deduction from equality to inequality. (Contributed by NM, 23-Feb-2005.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Hypothesis

Ref	Expression
necon3bid.1	|- ( ph -> ( A = B <-> C = D ) )

Assertion

Ref	Expression
necon3bid	|- ( ph -> ( A =/= B <-> C =/= D ) )

Proof of Theorem necon3bid

Step	Hyp	Ref	Expression
1		df-ne 2206	. 2 |- ( A =/= B <-> -. A = B )
2		necon3bid.1	. . 3 |- ( ph -> ( A = B <-> C = D ) )
3	2	necon3bbid 2245	. 2 |- ( ph -> ( -. A = B <-> C =/= D ) )
4	1, 3	syl5bb 181	1 |- ( ph -> ( A =/= B <-> C =/= D ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 98   = wceq 1243   =/= wne 2204

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545

This theorem depends on definitions:  df-bi 110  df-ne 2206

This theorem is referenced by:  nebidc  2285  addneintrd  7199  addneintr2d  7200  negne0bd  7315  negned  7319  subne0d  7331  subne0ad  7333  subneintrd  7366  subneintr2d  7368  qapne  8574  xrlttri3  8718  sqne0  9319  cjne0  9508  absne0d  9783