Theorem mtbi 681

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 133	. 2 |- ( ps -> ph )
4	1, 3	mto 672	1 |- -. ps

This proof depends on syntax axioms:   -. wn 3   <-> wb 105

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624

This proof depends on definitions:  df-bi 117

This theorem is used by:  mtbir  682  vnex  4264  onsucelsucexmid  4677  dtruex  4706  dmsn0  5255  php5  7159  exmidonfinlem  7545  ndvdsi  12716  nprmi  12918  dec2dvds  13210  dec5dvds2  13212  ballotfilem2  13277  unennn  13337  ppi2i  16178  bj-vprc  17020  bj-vnex  17022  trirec0xor  17192