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

Syntax hints:   -. wn 3   <-> 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 623  ax-in2 624

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  mtbir  682  vnex  4259  onsucelsucexmid  4672  dtruex  4701  dmsn0  5250  php5  7149  exmidonfinlem  7535  ndvdsi  12678  nprmi  12880  dec2dvds  13168  dec5dvds2  13170  ballotfilem2  13206  unennn  13266  bj-vprc  16836  bj-vnex  16838  trirec0xor  16999