Theorem pm5.21ni 676

Description: Two propositions implying a false one are equivalent.

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		pm5.21 675	. 2 |- ((-. ph /\ -. ch) -> (ph <-> ch))
2		pm5.21ni.1	. . 3 |- (ph -> ps)
3	2	con3i 98	. 2 |- (-. ps -> -. ph)
4		pm5.21ni.2	. . 3 |- (ch -> ps)
5	4	con3i 98	. 2 |- (-. ps -> -. ch)
6	1, 3, 5	sylanc 471	1 |- (-. ps -> (ph <-> ch))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146

This theorem is referenced by:  pm5.21nii 677  pm5.54 681  niabn 757  ordsucelsuc 3063  ndmord 4036  breng 4357  brdomg 4358  r1pw 4658  r1pwcl 4659  alephsucdom 4852  elioo3g 6317  elfz2t 6404

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

Copyright terms: Public domain