Theorem mtbi 630

Description: An inference from a biconditional, related to modus tollens. (Contributed by NM, 15-Nov-1994.) (Proof shortened by Wolf Lammen, 25-Oct-2012.)

Hypotheses

Ref	Expression
mtbi.1	|- -. ph
mtbi.2	|- ( ph <-> ps )

Assertion

Ref	Expression
mtbi	|- -. ps

Proof of Theorem mtbi

Step	Hyp	Ref	Expression
1		mtbi.1	. 2 |- -. ph
2		mtbi.2	. . 3 |- ( ph <-> ps )
3	2	biimpri 131	. 2 |- ( ps -> ph )
4	1, 3	mto 623	1 |- -. ps

Syntax hints:   -. wn 3   <-> wb 103

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580

This theorem depends on definitions:  df-bi 115

This theorem is referenced by:  mtbir  631  vnex  3962  onsucelsucexmid  4336  dtruex  4365  dmsn0  4885  php5  6554  ndvdsi  11015  nprmi  11188  unennn  11292  bj-vprc  11433  bj-vnex  11435