Theorem mtbi 627

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 131	. 2 ⊢ (𝜓 → 𝜑)
4	1, 3	mto 620	1 ⊢ ¬ 𝜓

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 576  ax-in2 577

This theorem depends on definitions:  df-bi 115

This theorem is referenced by:  mtbir  628  vprc  3909  vnex  3911  onsucelsucexmid  4273  dtruex  4302  dmsn0  4808  php5  6344  ndvdsi  10333  nprmi  10506  bj-vprc  10687  bj-vnex  10689