Theorem necon1ai 1605

Description: Contrapositive inference for inequality.

Hypothesis

Ref	Expression
necon1ai.1	|- (-. ph -> A = B)

Assertion

Ref	Expression
necon1ai	|- (A =/= B -> ph)

Proof of Theorem necon1ai

Step	Hyp	Ref	Expression
1		df-ne 1584	. 2 |- (A =/= B <-> -. A = B)
2		necon1ai.1	. . 3 |- (-. ph -> A = B)
3	2	con1i 96	. 2 |- (-. A = B -> ph)
4	1, 3	sylbi 199	1 |- (A =/= B -> ph)

Syntax hints:  -. wn 2   -> wi 3   = wceq 954   =/= wne 1582

This theorem is referenced by:  necon1i 1607  tz6.12i 3732  elfvdm 3738  1re 5415  ioossicc 6338

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-ne 1584

Copyright terms: Public domain