Theorem necon2bi 2455

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 2421	. 2 |- ( ph -> -. A = B )
3	2	con2i 630	1 |- ( A = B -> -. ph )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1395   =/= wne 2400

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-in1 617  ax-in2 618

This theorem depends on definitions:  df-bi 117  df-ne 2401

This theorem is referenced by:  minel  3553  rzal  3589  difsnb  3810  fin0  7043  0npi  7496  0nsr  7932  renfdisj  8202  nltpnft  10006  ngtmnft  10009  xrrebnd  10011  hashnncl  11012  rennim  11508  pceq0  12840