Theorem necon2bi 2458

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 2424	. 2 |- ( ph -> -. A = B )
3	2	con2i 632	1 |- ( A = B -> -. ph )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1398   =/= wne 2403

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-in1 619  ax-in2 620

This theorem depends on definitions:  df-bi 117  df-ne 2404

This theorem is referenced by:  minel  3558  rzal  3594  difsnb  3821  fin0  7117  0npi  7576  0nsr  8012  renfdisj  8281  nltpnft  10093  ngtmnft  10096  xrrebnd  10098  hashnncl  11103  rennim  11625  pceq0  12958