Theorem pm5.21ni 341

Description: Two propositions implying a false one are equivalent. (Contributed by NM, 16-Feb-1996.) (Proof shortened by Wolf Lammen, 19-May-2013.)

Hypotheses

Ref	Expression
pm5.21ni.1	|- (ph -> ps)
pm5.21ni.2	|- (ch -> ps)

Assertion

Ref	Expression
pm5.21ni	|- (-. ps -> (ph <-> ch))

Proof of Theorem pm5.21ni

Step	Hyp	Ref	Expression
1		pm5.21ni.1	. . 3 |- (ph -> ps)
2	1	con3i 127	. 2 |- (-. ps -> -. ph)
3		pm5.21ni.2	. . 3 |- (ch -> ps)
4	3	con3i 127	. 2 |- (-. ps -> -. ch)
5	2, 4	2falsed 340	1 |- (-. ps -> (ph <-> ch))

This proof depends on syntax axioms:  -. wn 3   -> wi 4   <-> wb 176

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

This theorem is used by:  pm5.21nii  342  pm5.54  870  niabn  917  ndmovord  5621