Theorem necon2bi 2402

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 2368	. 2 |- ( ph -> -. A = B )
3	2	con2i 627	1 |- ( A = B -> -. ph )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1353   =/= wne 2347

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-in1 614  ax-in2 615

This theorem depends on definitions:  df-bi 117  df-ne 2348

This theorem is referenced by:  minel  3486  rzal  3522  difsnb  3737  fin0  6888  0npi  7315  0nsr  7751  renfdisj  8020  nltpnft  9817  ngtmnft  9820  xrrebnd  9822  hashnncl  10778  rennim  11014  pceq0  12324