Theorem bibif 698

Description: Transfer negation via an equivalence. (Contributed by NM, 3-Oct-2007.) (Proof shortened by Wolf Lammen, 28-Jan-2013.)

Assertion

Ref	Expression
bibif	|- ( -. ps -> ( ( ph <-> ps ) <-> -. ph ) )

Proof of Theorem bibif

Step	Hyp	Ref	Expression
1		nbn2 697	. 2 |- ( -. ps -> ( -. ph <-> ( ps <-> ph ) ) )
2		bicom 140	. 2 |- ( ( ps <-> ph ) <-> ( ph <-> ps ) )
3	1, 2	bitr2di 197	1 |- ( -. ps -> ( ( ph <-> ps ) <-> -. ph ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 105

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  nbn  699