Theorem necon2bi 2260

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 2226	. 2 |- ( ph -> -. A = B )
3	2	con2i 557	1 |- ( A = B -> -. ph )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1243   =/= wne 2204

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-in1 544  ax-in2 545

This theorem depends on definitions:  df-bi 110  df-ne 2206

This theorem is referenced by:  minel  3283  rzal  3318  difsnb  3506  fin0  6342  0npi  6411  0nsr  6834  renfdisj  7079  nltpnft  8730  ngtmnft  8731  xrrebnd  8732  rennim  9600