Theorem necon2bi 2475

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 2441	. 2 |- ( ph -> -. A = B )
3	2	con2i 636	1 |- ( A = B -> -. ph )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   = wceq 1402   =/= wne 2420

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-in1 623  ax-in2 624

This proof depends on definitions:  df-bi 117  df-ne 2421

This theorem is used by:  minel  3586  rzal  3625  difsnb  3858  fin0  7189  0npi  7680  0nsr  8116  renfdisj  8385  nltpnft  10216  ngtmnft  10219  xrrebnd  10221  hashnncl  11234  rennim  11768  pceq0  13101