Theorem necon2bi 2382

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 2348	. 2 |- ( ph -> -. A = B )
3	2	con2i 617	1 |- ( A = B -> -. ph )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1335   =/= wne 2327

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-in1 604  ax-in2 605

This theorem depends on definitions:  df-bi 116  df-ne 2328

This theorem is referenced by:  minel  3455  rzal  3491  difsnb  3699  fin0  6830  0npi  7233  0nsr  7669  renfdisj  7937  nltpnft  9718  ngtmnft  9721  xrrebnd  9723  hashnncl  10670  rennim  10902