Theorem mtbi 660

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	⊢ ¬ 𝜑
mtbi.2	⊢ (𝜑 ↔ 𝜓)

Assertion

Ref	Expression
mtbi	⊢ ¬ 𝜓

Proof of Theorem mtbi

Step	Hyp	Ref	Expression
1		mtbi.1	. 2 ⊢ ¬ 𝜑
2		mtbi.2	. . 3 ⊢ (𝜑 ↔ 𝜓)
3	2	biimpri 132	. 2 ⊢ (𝜓 → 𝜑)
4	1, 3	mto 652	1 ⊢ ¬ 𝜓

Syntax hints:  ¬ wn 3   ↔ wb 104

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  mtbir  661  vnex  4113  onsucelsucexmid  4507  dtruex  4536  dmsn0  5071  php5  6824  exmidonfinlem  7149  ndvdsi  11870  nprmi  12056  unennn  12330  bj-vprc  13778  bj-vnex  13780  trirec0xor  13924