Theorem necon2abid 1622

Description: Contrapositive deduction for inequality.

Hypothesis

Ref	Expression
necon2abid.1	|- (ph -> (A = B <-> -. ps))

Assertion

Ref	Expression
necon2abid	|- (ph -> (ps <-> A =/= B))

Proof of Theorem necon2abid

Step	Hyp	Ref	Expression
1		necon2abid.1	. . 3 |- (ph -> (A = B <-> -. ps))
2	1	con2bid 526	. 2 |- (ph -> (ps <-> -. A = B))
3		df-ne 1587	. 2 |- (A =/= B <-> -. A = B)
4	2, 3	syl6bbr 538	1 |- (ph -> (ps <-> A =/= B))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   = wceq 956   =/= wne 1585

This theorem is referenced by:  leltnet 5521  xrleltnet 5558  xrltnet 5565  supxrbnd 6091  supxrre2 6094  nmoreltpnf 8432  nmopreltpnf 9796

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7

This theorem depends on definitions:  df-bi 147  df-an 225  df-ne 1587

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