Theorem necon2bi 2422

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 2388	. 2 |- ( ph -> -. A = B )
3	2	con2i 628	1 |- ( A = B -> -. ph )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1364   =/= wne 2367

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-in1 615  ax-in2 616

This theorem depends on definitions:  df-bi 117  df-ne 2368

This theorem is referenced by:  minel  3512  rzal  3548  difsnb  3765  fin0  6946  0npi  7380  0nsr  7816  renfdisj  8086  nltpnft  9889  ngtmnft  9892  xrrebnd  9894  hashnncl  10887  rennim  11167  pceq0  12491