Theorem necon3bi 2558

Description: Contrapositive inference for inequality. (Contributed by NM, 1-Jun-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Hypothesis

Ref	Expression
necon3bi.1	|- (A = B -> ph)

Assertion

Ref	Expression
necon3bi	|- (-. ph -> A =/= B)

Proof of Theorem necon3bi

Step	Hyp	Ref	Expression
1		nne 2521	. . 3 |- (-. A =/= B <-> A = B)
2		necon3bi.1	. . 3 |- (A = B -> ph)
3	1, 2	sylbi 187	. 2 |- (-. A =/= B -> ph)
4	3	con1i 121	1 |- (-. ph -> A =/= B)

This proof depends on syntax axioms:  -. wn 3   -> wi 4   = wceq 1642   =/= wne 2517

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 177  df-ne 2519

This theorem is used by:  r19.2zb  3641  nchoicelem8  6297