Theorem necon2bi 2301

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 2267	. 2 |- ( ph -> -. A = B )
3	2	con2i 590	1 |- ( A = B -> -. ph )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1285   =/= wne 2246

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-in1 577  ax-in2 578

This theorem depends on definitions:  df-bi 115  df-ne 2247

This theorem is referenced by:  minel  3312  rzal  3346  difsnb  3536  fin0  6419  0npi  6565  0nsr  6988  renfdisj  7239  nltpnft  8960  ngtmnft  8961  xrrebnd  8962  sizenncl  9820  rennim  10026