Theorem mtbi 665

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 657	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 609  ax-in2 610

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  mtbir  666  vnex  4120  onsucelsucexmid  4514  dtruex  4543  dmsn0  5078  php5  6836  exmidonfinlem  7170  ndvdsi  11892  nprmi  12078  unennn  12352  bj-vprc  13931  bj-vnex  13933  trirec0xor  14077