Theorem necon1ad 2584

Description: Contrapositive deduction for inequality. (Contributed by NM, 2-Apr-2007.)

Hypothesis

Ref	Expression
necon1ad.1	|- (ph -> (-. ps -> A = B))

Assertion

Ref	Expression
necon1ad	|- (ph -> (A =/= B -> ps))

Proof of Theorem necon1ad

Step	Hyp	Ref	Expression
1		df-ne 2519	. 2 |- (A =/= B <-> -. A = B)
2		necon1ad.1	. . 3 |- (ph -> (-. ps -> A = B))
3	2	con1d 116	. 2 |- (ph -> (-. A = B -> ps))
4	1, 3	syl5bi 208	1 |- (ph -> (A =/= B -> ps))

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:  nchoicelem15  6304