Theorem necon2bi 2475

Description: Contrapositive inference for inequality. (Contributed by NM, 1-Apr-2007.)

Hypothesis

Ref	Expression
necon2bi.1	|- ( ph -> A =/= B )

Assertion

Ref	Expression
necon2bi	|- ( A = B -> -. ph )

Proof of Theorem necon2bi

Step	Hyp	Ref	Expression
1		necon2bi.1	. . 3 |- ( ph -> A =/= B )
2	1	neneqd 2441	. 2 |- ( ph -> -. A = B )
3	2	con2i 636	1 |- ( A = B -> -. ph )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1402   =/= wne 2420

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-in1 623  ax-in2 624

This theorem depends on definitions:  df-bi 117  df-ne 2421

This theorem is referenced by:  minel  3585  rzal  3622  difsnb  3853  fin0  7179  0npi  7670  0nsr  8106  renfdisj  8375  nltpnft  10195  ngtmnft  10198  xrrebnd  10200  hashnncl  11212  rennim  11746  pceq0  13079